diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzekpi" "b/data_all_eng_slimpj/shuffled/split2/finalzzekpi" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzekpi" @@ -0,0 +1,5 @@ +{"text":"\\section{\\bf Introduction}\\label{section-1}\n\nThis paper is concerned with large-scale regularity estimates in the homogenization of elliptic systems of elasticity with \nperiodic high-contrast coefficients.\nLet $\\omega$ be a connected and unbounded open set in $\\mathbb{R}^d$. Assume that $\\omega$ is 1-periodic; i.e.,\nits characteristic function is periodic with respect to $\\mathbb{Z}^d$.\nWe also assume that \neach of connected components of\n$\n\\mathbb{R}^d\\setminus \\omega \n$\nis the closure of a bounded open set $F_k$ with Lipschitz boundary,\nand that\n\\begin{equation}\\label{dis}\n\\min_{k\\neq \\ell} \\text{dist} (F_k, F_\\ell ) >0.\n\\end{equation}\nFor $0< \\delta < \\infty$, define\n\\begin{equation}\\label{Lambda-e}\n\\Lambda_{\\delta} (x)\n=\\left\\{\n\\aligned\n& \\delta & \\quad & \\text{ if } x\\in F=\\cup_k F_k ,\\\\\n& 1 & \\quad & \\text{ if } x\\notin F.\n\\endaligned\n\\right.\n\\end{equation}\nWe are interested in the large-scale regularity estimates, that are uniform in $\\delta>0$, for the elliptic operator \n\\begin{equation}\\label{op}\n\\mathcal{L}_\\delta \n=-\\text{\\rm div} \\big(\\Lambda_{\\delta^2} A\\nabla \\big).\n\\end{equation}\nHere and thereafter the coefficient matrix (tensor) $A=A(x)= (a_{ij}^{\\alpha\\beta} (x))$,\nwith $1\\le \\alpha, \\beta, i, j \\le d$, is assumed to be real, \nbounded measurable, 1-periodic, and to satisfy the elasticity condition,\n\\begin{equation}\\label{ellipticity}\n\\aligned\n& a_{ij}^{\\alpha\\beta} (x) = a_{ji}^{\\beta\\alpha} (x) = a_{\\alpha j}^{i\\beta} (x) ,\\\\\n& \\kappa_1 |\\xi|^2 \\le a_{ij}^{\\alpha\\beta} \\xi_i^\\alpha \\xi_j^\\beta \\le \\kappa_2 |\\xi|^2\n\\endaligned\n\\end{equation}\nfor any symmetric matrix $\\xi= (\\xi_i^\\alpha)\\in \\mathbb{R}^{d\\times d}$,\nwhere $\\kappa_1$, $\\kappa_2$ are positive constants.\nUnder these assumptions we will show that if $u\\in H^1(Q_R; \\mathbb{R}^d)$ is a weak solution of\n$\\mathcal{L}_\\delta (u)=0$ in a cube $Q_R= (-R\/2, R\/2)^d$ of size $R$ for some $R\\ge 4$, then\n\\begin{equation}\\label{Lip-e}\n\\sup_{1\\le r\\le R-3}\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R\\cap \\omega} |\\nabla u|^2 \\right)^{1\/2},\n\\end{equation}\nwith a constant $C$ independent of $R$ and $\\delta$.\nLet $ Du $ denote the symmetric gradient of $u$; i.e., \n$$\nDu = \\big( \\nabla u + (\\nabla u)^T\\big)\/2,\n$$\n where\n$(\\nabla u)^T$ denotes the transpose of $\\nabla u$. We also prove that for $R\\ge 4$,\n\\begin{equation}\\label{Lip-e-1}\n\\sup_{1\\le r\\le R-3}\n\\left(\\fint_{Q_r } |D u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R\\cap \\omega } |D u|^2 \\right)^{1\/2}.\n\\end{equation}\nWe remark that the operator $\\mathcal{L}_{\\delta}$ arises naturally in the modeling of acoustic propagations in porous media, diffusion processes in highly heterogeneous media, and\n inclusions in composite materials \\cite{Arbogast-1990, Allaire-1992, OSY-1992, JKO-1993}.\n\nIn the case $\\delta=1$,\nthe regularity estimates for the elliptic system $-\\text{\\rm div}(A(x\/\\varepsilon)\\nabla)=f$\nin the homogenization theory \nhave been studied extensively in recent years (in this paper we have rescaled the equation so that the microscopic scale $\\varepsilon=1$ and the domain is large).\nUsing a compactness method,\nthe interior Lipschitz estimate and the boundary Lipschitz estimate for the Dirichlet problem in\na $C^{1, \\alpha}$ domain were established by M. Avellaneda and F. Lin in a seminal work \\cite{AL-1987}.\nThe boundary Lipschitz estimate for the Neumann problem in a $C^{1, \\alpha}$ domain \nwas obtained in \\cite{KLS-2013-N}.\nWe refer the reader to \\cite{Shen-book} for further references on periodic homogenization, and to \\cite{Armstrong-book}\nfor related work on the large-scale regularity in stochastic homogenization.\n\nIn this paper we will be concerned with the case $\\delta \\neq 1$,\nwhere, in the simpler scalar case,\n$\\delta^2$ represents the conductivity ratio (or the ratio of diffusion coefficients) of the disconnected {\\it inclusions} $F=\\cup_k F_k$ to the connected \\text{\\it matrix} $\\omega$.\nNotice that the operator $\\mathcal{L}_{\\delta}$ is elliptic, but neither uniformly in $\\delta \\in (0, 1)$ nor in $\\delta \\in (1, \\infty)$.\n We mention that in the scalar case with $0\\le \\delta<1$, $A=I$ and \n $\\omega$ being sufficiently smooth, using the compactness method in \\cite{AL-1987}, \nthe $W^{1, p}$ and Lipschitz estimates were obtained by L.-M. Yeh \\cite{Yeh-2010, Yeh-2011, Yeh-2015, Yeh-2016}.\nAlso see earlier work in \\cite{Schweizer-2000, Masmoudi-2004} for related uniform estimates in the case $\\delta=0$.\nIn \\cite{Chase-Russell-2017} B. Russell established the large-scale interior Lipschitz estimate\nfor the system of elasticity with bounded measurable coefficients in the case $\\delta=0$, using an approximation method\noriginated in \\cite{Armstrong-Smart-2016}. \nThe case $0<\\delta<1$ was treated in \\cite{Chase-Russell-2018}.\nIn the stochastic setting with $\\delta=0$, S. Armstrong and P. Dario \\cite{Armstrong-2018} obtained \nquantitative homogenization and large-scale regularity results for the random conductance model on a supercritical percolation.\n\nThe following is one of the main results of this paper.\n\n\\begin{thm}\\label{main-thm-1}\nLet $0\\le \\delta\\le \\infty$.\nAssume that $A$ satisfies the elasticity condition \\eqref{ellipticity} and is 1-periodic.\nLet $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q_R$ for some $R\\ge 4$.\nThen \\eqref{Lip-e} and \\eqref{Lip-e-1}\nhold for some constant $C$ depending only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{thm}\n\nNote that Theorem \\ref{main-thm-1} includes the limiting cases of periodically perforated domains: $\\delta=0$ and $\\delta=\\infty$.\nIn the case $\\delta=0$, which is referred to as the soft inclusions \\cite{JKO-1993}, we call $u\\in H^1(\\Omega; \\mathbb{R}^d)$ \nis a weak solution of $\\mathcal{L}_0 (u)=f\\chi_\\omega $ in $\\Omega$, if \n$$\n\\int_{\\Omega\\cap \\omega} A\\nabla u \\cdot \\nabla v \\, dx =\\int_{\\Omega\\cap \\omega} f\\cdot v \\, dx\n$$\nfor any $ v \\in H_0^1(\\Omega; \\mathbb{R}^d)$.\nFormally, this means \nthat $ -\\text{\\rm div} (A\\nabla u)=f$ in $\\Omega \\cap \\omega$ and $\\big(\\frac{\\partial u}{\\partial \\nu }\\big)_-=0$ on \n$\\Omega \\cap \\partial \\omega$, where \n$\n \\big( \\frac{\\partial u}{\\partial \\nu } \\big)_- = n \\cdot A (\\nabla u )_-\n$ denotes the conormal derivative taken from $\\omega$ and \n$n$ the outward unit normal to $\\partial F$.\nFor convenience we will also assume that $u$ is a weak solution of\n$\\text{\\rm div} (A\\nabla u)=0$ in $\\Omega \\cap F$.\nIn the case $\\delta=\\infty$, which is referred to as the stiff inclusions \\cite{JKO-1993}, \n a function $u$ in $H^1(\\Omega; \\mathbb{R}^d)$ is called a weak solution of\n$\\mathcal{L}_\\infty (u)=f$ in $\\Omega$ if\n$ Du =0 \\text{ in }\\Omega \\cap F$, and \n$$\n\\int_{\\Omega\\cap \\omega} A\\nabla u \\cdot \\nabla v \\, dx =\\int_\\Omega f \\cdot v\\, dx\n$$\nfor any $ v \\in H_0^1(\\Omega; \\mathbb{R}^d)$ with\n$D v =0$ in $\\Omega \\cap F$.\nThis implies that $-\\text{\\rm div}(A\\nabla u)=f$ in $\\Omega\\cap \\omega$ and that if $\\overline{F}_k\\subset \\Omega$,\n$$\n\\int_{\\partial F_k} \\Big( \\frac{\\partial u}{\\partial \\nu}\\Big)_- \\cdot \\phi\\, d\\sigma =-\\int_{F_k} f \\cdot \\phi\\, dx\n$$\nfor any $\\phi\\in \\mathcal{R}$, the space of rigid displacements.\n\nThe large-scale uniform Lipschitz estimate in Theorem \\ref{main-thm-1}, which holds under the assumptions that $A$ is bounded measurable and\n$\\partial \\omega$ is locally Lipschitz, is new in the case $1<\\delta\\le \\infty$,\neven when $A$ is constant and $\\omega$ is smooth.\nUnder the additional conditions that $\\omega$ is locally $C^{1,\\alpha}$ and $A$ is H\\\"older continuous,\n\\begin{equation}\\label{smoothness}\n|A(x)-A(y)|\\le M_0 |x-y|^\\sigma \\quad \\text{ for any } x, y\\in \\mathbb{R}^d,\n\\end{equation} \nwhere $M_0> 0$ and $\\sigma \\in (0, 1)$,\nwe may combine \\eqref{Lip-e} with the local Lipschitz estimates for the operator $\\mathcal{L}_\\delta$\nto obtain a true Lipschitz estimate.\n\n\\begin{thm}\\label{main-thm-2}\nLet $0\\le \\delta\\le \\infty$ and $Q(x_0, R)=x_0 + Q_R$.\nAssume that $A$ satisfies conditions \\eqref{ellipticity}, \\eqref{smoothness}, and is 1-periodic.\nAlso assume that each $F_k$ is a bounded $C^{1, \\alpha}$ domain for some\n$\\alpha \\in (0, 1)$.\nLet $u\\in H^1(Q(x_0, R); \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in \n$Q(x_0, R) $ for some $R\\ge 4$.\nThen \n\\begin{equation}\\label{f-Lip}\n|\\nabla u(x_0)|\n\\le C \\left(\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$ in \\eqref{smoothness}.\n\\end{thm}\n\nThe Lipschitz estimate \\eqref{f-Lip} as well as its small-scale analogue allows\nus to construct a $d\\times d$ matrix $\\Gamma_\\delta (x, y)$ of fundamental solutions for the operator $\\mathcal{L}_\\delta$ in $\\mathbb{R}^d$, and \nobtain its estimates that are uniform in $\\delta \\in (0, \\infty)$.\nIn particular, we will show that if $d\\ge 3$ and $1\\le \\delta<\\infty$,\n\\begin{equation}\\label{fund-0}\n\\left\\{\n\\aligned\n |\\Gamma_\\delta (x, y)| & \\le C |x-y|^{2-d},\\\\\n |\\nabla_x \\Gamma_\\delta (x,y)|\n +|\\nabla_y \\Gamma_\\delta(x, y) |\n & \\le C |x-y|^{1-d},\\\\\n |\\nabla_x \\nabla_y \\Gamma_\\delta (x, y)| & \\le C |x-y|^{-d}\n \\endaligned\n \\right.\n \\end{equation}\n for any $x, y\\in \\mathbb{R}^d$ and $x\\neq y$,\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$.\n In the case $0<\\delta<1$, the estimates in \\eqref{fund-0} continue to hold, provided that either \n $|x-y|_\\infty \\ge 4$ or $x, y\\in \\omega$.\n Here $|x-y|_\\infty =\\max ( |x_1-y_1|, \\dots, |x_d -y_d|)$ denotes the $L^\\infty$ norm in $\\mathbb{R}^d$.\n See Theorems \\ref{thm-f-1}, \\ref{thm-f-1a} and \\ref{thm-f-3}.\n We mention that in the scalar case with $A=I $ and $0<\\delta<1$, explicit bounds for fundamental solutions were obtained by L.-M. Yeh in \\cite{Yeh-2016}.\n As in the case $\\delta=1$ \\cite{AL-1991, KS-2011-L, KLS-2014},\n estimates of fundamental solutions are an important tool in the study of optimal regularity problems in the homogenization theory for solutions\n of $\\mathcal{L}_\\delta (u)=f$.\n In particular, it allows us to extend the Lipschitz estimate \\eqref{f-Lip}\n from solutions of $\\mathcal{L}_\\delta (u)=0$ to that of $\\mathcal{L}_\\delta (u)=f$.\n Indeed, under the same assumptions on $A$ and $\\omega$ as in Theorem \\ref{main-thm-2},\n we obtain \n \\begin{equation}\\label{f-Lip-f}\n|\\nabla u(x_0)|\n\\le C_p \\left\\{ \\left(\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2}\n+R \\left(\\fint_{Q(x_0, R)} |f|^p \\right)^{1\/p} \\right\\}\n\\end{equation}\nfor $1\\le \\delta\\le \\infty$,\nwhere $u$ is a weak solution of $\\mathcal{L}_\\delta (u)=f$ in $Q(x_0, R) $ for some\n$R\\ge 4$ and $p>d$.\nIf $0\\le \\delta<1$, the estimate (\\ref{f-Lip-f}) holds for solutions of $\\mathcal{L}_\\delta (u)=f \\chi_\\omega$ in $Q(x_0, R)$.\nSee Theorem \\ref{main-thm-3}.\n \n We now describe our general approach to the proof of Theorem \\ref{main-thm-1}.\n As we mentioned earlier,\n the scalar case with $0\\le \\delta<1$ and $A=I$ was studied in \\cite{Yeh-2010, Yeh-2011, Yeh-2015, Yeh-2016},\n using a compactness method of Avellaneda and Lin \\cite{AL-1987}.\n The compactness argument is fairly complicated to implement for the operator $\\mathcal{L}_\\delta$,\n as both the coefficient matrix $A$ and the ratio $\\delta^2$ should be allowed to vary.\n A more direct approach, which originated in \\cite{Armstrong-Smart-2016}, was used in \\cite{Chase-Russell-2017, Chase-Russell-2018}\n to treat the case $0\\le \\delta<1$ with bounded measurable coefficients.\n The approach relies on a result on the convergence rate, uniform in $\\delta$, for the operator $-\\text{\\rm div}(\\Lambda_{\\delta^2} A(x\/\\varepsilon) \\nabla )$\n as $\\varepsilon\\to 0$.\n It is not clear how to extend either of these two methods to the case $1<\\delta\\le \\infty$.\n In this paper we will adapt a more recent method of S. Armstrong, T. Kuusi, and C. Smart \\cite{Armstrong-2020},\n which is based on a Caccioppoli type inequality and the fact that $\\Delta_j u$ is a solution whenever $u$ is a solution,\n where $\\Delta_j$ denotes the difference operator,\n \\begin{equation}\\label{diff}\n \\Delta_j u (x) = u(x+e_j) -u(x)\n \\end{equation}\n for $1\\le j\\le d$ and $e_j= (0, \\dots, 1, \\dots, 0)$ with $1$ in the $j^{\\rm th}$ place.\n The basic idea is to transfer the higher-order regularity of $u$ in terms of the difference operator to \n higher-order regularity of $u$ at a large scale through Caccioppoli and Poincar\\'e's inequalities.\n For elliptic systems the approach also uses a discrete Sobolev inequality.\n \n To carry out the approach described above, a key step is to establish a Caccioppoli inequality for solutions of $\\mathcal{L}_\\delta (u)=0$\n in $Q_R$ for $R$ large. In the case $0\\le \\delta\\le 1$, it can be shown by an extension argument that \n\\begin{equation}\\label{C-0}\n\\int_{Q_{R\/2}} |\\nabla u|^2\\, dx \\le \\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx,\n\\end{equation}\nwhich is more or less known \\cite{Chase-Russell-2017, Chase-Russell-2018}.\nIt is not known that \\eqref{C-0} holds for the case $1< \\delta\\le \\infty$, with constant $C$ independent of $\\delta$.\nHowever, if $\\delta$ is sufficiently large or $\\delta=\\infty$,\nwe are able to show that for any $\\ell\\ge 1$ and $R\\ge 32$,\n\\begin{equation}\\label{C-1}\n\\int_{Q_{R\/2 }} |\\nabla u|^2\\, dx \\le \\frac{C_\\ell }{R^2} \\int_{Q_R} |u|^2\\, dx + \\frac{C_\\ell}{R^{2\\ell } }\n\\int_{Q_R} |\\nabla u|^2\\, dx,\n\\end{equation}\nby some extension and iteration arguments.\nIt turns out that the weaker version \\eqref{C-1} with $\\ell=1$, together with the discrete Sobolev inequality, is sufficient to\ncomplete the proof of \\eqref{Lip-e}.\nWe point out that the method described above does not extend to the nonhomogeneous system $\\mathcal{L}_\\delta (u) =f$\nwith nonsmooth $f$. \nWe resolve this issue by introducing the matrix of fundamental solutions.\n\nThe paper is organized as follows. In Section \\ref{section-2} we give the proof of \\eqref{C-0}.\nThe inequality \\eqref{C-1} is proved in Section \\ref{section-3}, while the proof of Theorem \\ref{main-thm-1} is given in Section \\ref{section-4}.\nIn Section \\ref{section-5} we collect some known results on local estimates and give the proof of Theorem \\ref{main-thm-2}.\nThe matrix of fundamental solutions is introduced and studied in Section \\ref{section-6}.\nFinally, we establish the Lipschitz estimate for solutions of $\\mathcal{L}_\\delta (u)=f$ in Section \\ref{section-7}.\n\nRecall that $Q_R =(-R\/2, R\/2)^d$ and $Q(x_0, R)=x_0 + Q_R$ for $R>0$ and $x_0\\in \\mathbb{R}^d$.\nWe use $\\fint_E u=\\frac{1}{|E|} \\int_E u$ to denote the $L^1$ average of $u$ over a set $E$.\nWe use $C$ to denote a positive constant that may depend on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\nIf $C$ depends also on other parameters, it will be stated explicitly.\nWe emphasize that the results in Sections 2 - 4 hold with no smoothness condition on $A$ or $F=\\mathbb{R}^d \\setminus \\overline{\\omega}$ \nbeyond that $A$ is bounded measurable and $F$ is locally Lipschitz.\nIn Sections 5 - 7 we impose the H\\\"older continuity condition \\eqref{smoothness} on $A$ and also assume that $F$ is locally $C^{1, \\alpha}$.\n\n\n\\section{\\bf Preliminaries}\\label{section-2}\n\nThroughout this paper we assume that $\\omega$ is a connected, unbounded and 1-periodic open set in $\\mathbb{R}^d$.\nWrite\n\\begin{equation}\\label{F1}\n\\mathbb{R}^d\\setminus \\omega =\\cup_k \\overline{F}_k, \n\\end{equation}\nwhere each $\\overline{F}_k$ is the closure of a bounded Lipschitz domain $F_k$ with connected boundary.\nWe assume that $\\{ \\overline{F}_k \\} $ are mutually disjoint and satisfy the condition (\\ref{dis}).\nThis allows us to construct a sequence of mutually disjoint open sets $\\{\\widetilde{F}_k \\}$ with connected smooth boundary such that $\\overline{F}_k \\subset \\widetilde{F}_k$,\n\\begin{equation}\\label{dist}\n\\left\\{\n\\aligned\n& c_0 \\le \\text{dist}(\\partial F_k, \\partial \\widetilde{F}_k), \\\\\n& c_0 \\le \\text{dist} (\\widetilde{F}_k, \\widetilde{F}_\\ell ) \\text{ for } k\\neq \\ell,\n\\endaligned\n\\right.\n\\end{equation}\nfor some $c_0>0$.\nNote that by the periodicity of $\\omega$, \n$\\{ F_k\\}$ are the shifts of a finite number of bounded Lipschitz domains contained in $Q_2$.\nAs a result, \nwe may assume that $\\{ \\widetilde{F}_k \\} $ are the shifts of a finite number of bounded smooth domains contained in $Q_{5\/2}$.\n\n Let $\\mathcal{R}$ denote the space of rigid displacements of $\\mathbb{R}^d$; i.e.,\n \\begin{equation}\\label{R}\n \\mathcal{R}\n =\\big\\{ u = E + Bx: \\ E\\in \\mathbb{R}^d \\text{ and } B^T=-B \\big\\},\n \\end{equation}\n where $B^T$ denotes the transpose of the $d\\times d$ matrix $B$.\n The following extension lemma will be useful for us.\n\n\\begin{lemma}\\label{lemma-ex}\nLet $F_k$ and $\\widetilde{F}_k$ be given above.\nThere exists a linear extension operator \n$$\nP_k: H^1(\\widetilde{F}_k \\setminus \\overline{F}_k; \\mathbb{R}^d)\n\\to H^1(\\widetilde{F}_k; \\mathbb{R}^d)\n$$ \nsuch that\n\\begin{align}\n & P_k (u)=u \\quad \\text{ for any } u \\in \\mathcal{R}, \\label{ex-1}\\\\\n & \\| P_k (u)\\|_{H^1(\\widetilde{F}_k)} \n \\le C \\big( \\| u \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n + \\| Du \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)} \\big), \\label{ex-2}\\\\\n & \\|\\nabla P_k(u) \\|_{L^2(\\widetilde{F}_k)}\n \\le C \\| \\nabla u\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}, \\label{ex-3}\\\\\n & \\| D P_k (u) ) \\|_{L^2(\\widetilde{F}_k)}\n \\le C \\| D u \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}, \\label{ex-4}\n\\end{align}\nwhere $Du$ denotes the symmetric gradient of $u$ and $C$ depends only on $d$ and $\\omega$.\n\\end{lemma}\n\n\\begin{proof} \nSee \\cite[pp.45-47]{OSY-1992}.\nNote that since $\\widetilde{F}_k$ and $F_k$ are shifts of a finite number of domains,\nthe constant $C$ does not depend on $k$.\n\\end{proof}\n\nThroughout the paper we assume that $A$ is real, bounded measurable, 1-periodic, and satisfies the elasticity condition \\eqref{ellipticity}.\nIt is well known that \\eqref{ellipticity} implies \n\\begin{align}\n A\\xi \\cdot \\zeta & \\le \\frac{\\kappa_2}{4} |\\xi +\\xi^T| | \\zeta +\\zeta^T| \\label{e-1},\\\\\n \\frac{\\kappa_1}{4} \n |\\xi +\\xi^T| & \\le A \\xi \\cdot \\xi \\label{e-2}\n \\end{align}\n for any $d\\times d$ matrices $\\xi$ and $\\zeta$ \\cite[pp.30-31]{OSY-1992}.\n \n\\begin{lemma}\\label{lemma-2.1}\nLet $0<\\delta <\\infty$ and $u\\in H^1(\\Omega; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)=f$ in $\\Omega$.\nThen \n\\begin{equation}\\label{Ca-2.1}\n\\int_{\\Omega} \n|\\Lambda_\\delta Du |^2 | \\varphi|^2 \\, dx\n\\le C \\int_{\\Omega} |\\Lambda_\\delta u|^2 |\\nabla \\varphi |^2\\, dx\n+ C \\int_\\Omega |f | |u| |\\varphi|^2\\, dx\n\\end{equation}\nfor any $\\varphi \\in C_0^1(\\Omega)$,\nwhere $C$ depends only $d$, $\\kappa_1$ and $\\kappa_2$.\nIn the case $\\delta=0$, \\eqref{Ca-2.1} holds for solutions of $\\mathcal{L}_0 (u)=f\\chi_\\omega$ in $\\Omega$.\n\\end{lemma}\n\n\\begin{proof}\nAssume $0<\\delta< \\infty$.\nLet $v =u\\varphi^2$, where $\\varphi \\in C_0^1(\\Omega)$.\nSince\n$$\n\\int_{\\Omega} \\Lambda_{\\delta^2} A\\nabla u \\cdot \\nabla v \\, dx =\\int_\\Omega f \\cdot v\\, dx,\n$$\nwe see that \n$$\n\\int_{\\Omega}\n\\Lambda_{\\delta^2} ( A\\nabla u \\cdot \\nabla u) \\varphi^2\\, dx\n=-2 \\int_{\\Omega} \\Lambda_{\\delta^2} (A\\nabla u \\cdot u (\\nabla \\varphi) ) \\varphi\\, dx\n+\\int_\\Omega f \\cdot v\\, dx,\n$$\nfrom which the inequality \\eqref{Ca-2.1} follows by using \\eqref{e-1}-\\eqref{e-2} and the Cauchy inequality.\nThe fact that $|A\\nabla u|\\le C |D u|$ is also needed.\nThe case $\\delta=0$ may be handled in the same manner.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma-2.2}\nLet $u\\in H^1(\\widetilde{F}_k; \\mathbb{R}^d)$ be a weak solution of $-\\text{\\rm div}(A\\nabla u)=f$ in $F_k$.\nThen\n\\begin{align}\n\\int_{F_k} |\\nabla u|^2\\, dx & \\le C \\int_{\\widetilde{F}_k \\setminus \\overline{ F}_k} |\\nabla u|^2\\, dx\n+ C \\int_{{F}_k} |f|^2\\, dx \\label{2.2-0},\\\\\n\\int_{F_k} | Du |^2\\, dx & \\le C \\int_{\\widetilde{F}_k \\setminus \\overline{F}_k} | Du |^2\\, dx \\label{2.2-1}\n+C \\int_{{F}_k} |f|^2\\, dx,\n\\end{align}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemma-ex} there exists $w\\in H^1(\\widetilde{F}_k; \\mathbb{R}^d)$ such that $w=u$ on $\\widetilde{F}_k \\setminus F_k$ and\n$$\n \\| w\\|_{H^1(\\widetilde{F}_k) } \\le C \\| u\\|_{H^1(\\widetilde{F}_k \\setminus \\overline{F}_k)}.\n $$\nSince $\\text{\\rm div} (A\\nabla (u-w))=f-\\text{\\rm div}(A\\nabla w)$ in $F_k$ and $u-w\\in H^1_0(F_k; \\mathbb{R}^d)$,\nby the classical energy estimate,\n$$\n\\aligned\n\\|\\nabla u\\|_{L^2(F_k)}\n & \\le C \\big\\{ \\| f\\|_{L^2(F_k)}\n+ \\| \\nabla w \\|_{L^2(F_k)} \\big\\}\\\\\n& \\le C \\big\\{ \\| f\\|_{L^2(F_k)} +\n\\| u\\|_{H^1(\\widetilde{F}_k\\setminus \\overline{F}_k)} \\big\\}.\n\\endaligned\n$$\nNote that for any $\\phi\\in \\mathcal{R}$, \n$u-\\phi$ satisfies the same condition as $u$.\nIt follows that \n\\begin{equation}\\label{2.2-2}\n\\|\\nabla u -\\nabla \\phi \\|_{L^2(F_k)}\n\\le C \\big\\{ \\| f\\|_{L^2(F_k)} + \\| u-\\phi \\|_{H^1(\\widetilde{F}_k\\setminus \\overline{F}_k)} \\big\\}.\n\\end{equation}\nBy taking $\\phi$ to be the $L^1$ average of $u$ over $\\widetilde{F}_k \\setminus \\overline{F}_k$ and\nusing Poincar\\'e's inequality we obtain \\eqref{2.2-0}.\nTo see \\eqref{2.2-1}, we use\n$$\n\\| D u \\|_{L^2(F_k)}\n\\le \\|\\nabla u -\\nabla \\phi \\|_{L^2(F_k)}\n\\le C\\big\\{ \\| f\\|_{L^2(F_k)} + \\| u-\\phi \\|_{H^1(\\widetilde{F}_k\\setminus \\overline{F}_k)} \\big\\}.\n$$\nSince this holds for any $\\phi\\in \\mathcal{R}$, \n \\eqref{2.2-1} follows by the second Korn inequality \\cite[p.19]{OSY-1992}.\n\\end{proof}\n\n\\begin{remark}\\label{remark-p}\nIt follows from Lemma \\ref{lemma-2.2} that if $\\mathcal{L}_\\delta (u) =0 $ in $Q_{R+3}$\nfor some $R>0$, then\n\\begin{equation}\\label{ex-1a}\n\\int_{Q_R} |\\nabla u|^2\\, dx\n\\le C \\int_{Q_{R+3}\\cap \\omega} \n |\\nabla u|^2\\, dx\n \\quad\n \\text{ and } \\quad\n \\int_{Q_R} | Du |^2\\, dx\n \\le C \\int_{Q_{R+3}\\cap \\omega} | Du |^2\\, dx.\n \\end{equation}\n To see this, it suffices to note that if $F_k \\cap Q_R\\neq \\emptyset$, then\n $\\widetilde{F}_k \\setminus \\overline{F} _k \\subset Q_{R+3}\\cap \\omega$.\n Also, observe that by Sobolev inequality, for any $u\\in H^1(\\widetilde{F}_k; \\mathbb{R}^d)$, \n \\begin{equation}\\label{ex-2a}\n \\int_{\\widetilde{F}_k} |u|^2\\, dx \n \\le C \\int_{\\widetilde{F}_k} |\\nabla u|^2\\, dx\n + C \\int_{\\widetilde{F}_k \\setminus \\overline{F}_k} |u|^2\\, dx.\n \\end{equation}\n This, together with \\eqref{2.2-0}, implies that if \n $\\mathcal{L}_\\delta (u)=0$ in $Q_{R+3}$ for some $R>0$, then\n \\begin{equation}\\label{ex-3a}\n \\int_{Q_R} |u|^2\\, dx\n \\le C \\int_{Q_{R+3}\\cap \\omega} \n |u|^2\\, dx\n + \\int_{Q_{R+3}\\cap \\omega} \n |\\nabla u|^2\\, dx.\n \\end{equation}\n\\end{remark}\n\nThe next theorem gives a Caccioppoli inequality, which is uniform in $\\delta \\in [0, 1]$,\n for $\\mathcal{L}_\\delta$.\n \n\\begin{thm}\\label{thm-2.3}\nSuppose $0\\le \\delta < \\infty $.\nLet $u\\in H^1(Q_{2R}; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)=0$ in $Q_{2R} $ for some $R\\ge 4$.\nThen\n\\begin{equation}\\label{Ca-1}\n\\int_{Q_{R}} |\\nabla u|^2\\, dx\n\\le \\frac{C(1+\\delta^2) }{R^2}\n\\int_{Q_{2R} } |u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{thm}\n\n\\begin{proof}\n\nBy the second Korn inequality,\n\\begin{equation}\\label{K-1}\n\\int_{Q_R} \n|\\nabla u|^2 \\le C \\int_{Q_R} | Du |^2\\, dx\n+\\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$.\nIn \\eqref{Ca-2.1} we choose $\\varphi\\in C_0^1(Q_{2R})$ such that $\\varphi=1$ in $Q_{R+3}$ and\n$|\\nabla \\varphi|\\le C\/R$. This gives\n\\begin{equation}\\label{2.3-1}\n\\int_{Q_{R+3} \\cap \\omega}\n|D u |^2\\, dx \\le \\frac{C(1+\\delta^2) }{R^2} \\int_{Q_{2R}} |u|^2\\, dx.\n\\end{equation}\nwhich, together with \\eqref{K-1} and \\eqref{ex-1a}, gives \\eqref{Ca-1}.\n\\end{proof}\n\n\n\\section{A Caccioppoli type inequality for $1< \\delta\\le \\infty$}\\label{section-3}\n\nWe first consider the case $1<\\delta<\\infty$.\n\n \\begin{lemma}\\label{lemma-7.1}\n Let $ 1 < \\delta< \\infty$.\n Let $u\\in H^1(\\widetilde{F}_k; \\mathbb{R}^d)$ be a weak solution of\n $-\\text{\\rm div}(\\Lambda_{\\delta^2} A\\nabla u)=f$ in $\\widetilde{F}_k$.\n Then\n \\begin{equation}\\label{7.1-0}\n \\delta^2 \\| Du\\|_{L^2(F_k)}\n \\le C \\big\\{ \\| f\\|_{L^p (\\widetilde{F}_k)}\n + \\| Du\\|_{L^2(\\widetilde{F}_k\\setminus \\overline{F}_k)} \\big\\},\n \\ \\end{equation}\n where $p=\\frac{2d}{d+2}$ for $d\\ge 3$ and $p>1$ for $d=2$.\nThe constant $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n \n \\begin{proof}\n Let $v\\in H_0^1(\\widetilde{F}_k; \\mathbb{R}^d)$ be an extension of $u$ from $F_k$ to \n $\\widetilde{F}_k$ such that \n \\begin{equation}\\label{7.1-1}\n\\|v\\|_{H^1(\\widetilde{F}_k)}\n \\le C \\| u\\|_{H^1(F_k)}.\n \\end{equation}\n Since\n $$\n \\int_{\\widetilde{F}_k \\setminus \\overline{F}_k}\n A\\nabla u \\cdot \\nabla v\\, dx\n +\\delta^2 \\int_{F_k} A\\nabla u \\cdot \\nabla v\\, dx=\\int_{\\widetilde{F}_k} f \\cdot v\\, dx,\n $$\n it follows that\n \\begin{equation}\\label{7.1-2}\n \\aligned\n \\delta^2\\int_{F_k} |D u|^2\\, dx\n & \\le C \\| f\\|_{L^p(\\widetilde{F}_k)} \\| v\\|_{L^{p^\\prime} (\\widetilde{F}_k)}\n + C \\| Du \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)} \\| D v \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\\\\\n & \\le C ( \\| f\\|_{L^p(\\widetilde{F}_k)} \n + \\| Du \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}) \n \\| u\\|_{H^1(F_k)},\n \\endaligned\n \\end{equation}\n where we have used Sobolev inequality and \\eqref{7.1-1}.\n We now choose $\\phi\\in \\mathcal{R}$ such that\n $$\n \\| u-\\phi \\|_{H^1(F_k)} \\le C \\| D u \\|_{L^2(F_k)}.\n $$\n Since $u-\\phi$ satisfies the same conditions as $u$, we may deduce \\eqref{7.1-0},\n readily from \\eqref{7.1-2}, with $u-\\phi$ in the place of $u$.\n \\end{proof}\n\n\\begin{lemma}\\label{lemma-3.2}\nSuppose $1< \\delta< \\infty$.\nLet $u\\in H^1(Q_{R}; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q_{R}$\nfor some $R\\ge 16$.\nThen, for $(R\/2)\\le r\\le R-8$ and $0<\\varepsilon<1$,\n\\begin{equation}\\label{3.2-0}\n\\int_{Q_r}\n|\\nabla u |^2\\, dx\n\\le \\frac{C}{\\varepsilon (R-r)^2} \\int_{Q_{R}} |u|^2\\, dx\n+ \\Big(\\varepsilon +\\frac{C}{\\delta^2} \\Big) \\int_{Q_{R}} |\\nabla u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{lemma}\n\n\\begin{proof}\nAs in the proof of Theorem \\ref{thm-2.3},\nit follows from the second Korn inequality and \\eqref{2.2-1} that\n\\begin{equation}\\label{3.2-1}\n\\int_{Q_r} |\\nabla u|^2\\, dx\n\\le C \\int_{Q_{r+3}\\cap \\omega} |Du |^2\\, dx\n+\\frac{C}{r^2} \\int_{Q_r} |u|^2\\, dx.\n\\end{equation}\nSince $r\\ge R-r$, it suffices to bound the first term in the right-hand side of \\eqref{3.2-1}.\nTo this end,\nlet $\\varphi$ be a function in $C_0^1(Q_{R-3})$ such that\n$\\varphi=1$ in $Q_{r+3}$ and \n$$\n|\\nabla \\varphi|\\le C(R-r-6)^{-1}\n\\le C (R-r)^{-1},\n$$\nwhere we have used the assumption $R-r \\ge 8$.\nRecall that if $F_k\\cap Q_{R-3}\\neq \\emptyset$, then $\\widetilde{F}_k\\subset Q_{R}$.\nFor each $F_k$ with $\\widetilde{F}_k\\subset Q_{R}$, we let $w_k\\in H_0^1(\\widetilde{F}_k; \\mathbb{R}^d)$\nbe an extension of $u\\varphi^2-g_k $ from $F_k$ to $\\widetilde{F}_k$ with the property that\n\\begin{equation}\\label{p-1}\n\\| w_k \\|_{H^1(\\widetilde{F}_k)} \\le C \\| u\\varphi^2 -g_k \\|_{H^1(F_k)},\n\\end{equation}\nwhere $ g_k \\in \\mathcal{R} $ is to be determined.\nExtend $w_k$ from $\\widetilde{F}_k $ to $\\mathbb{R}^d $ by zero and let\n\\begin{equation}\\label{p-3}\n\\phi=u\\varphi^2 -\\sum_k w_k \\quad \\text{ in } \\mathbb{R}^d,\n\\end{equation}\nwhere the sum is taken over those $k$'s for which $\\widetilde{F}_k \\subset Q_R$.\nNote that $\\phi (x) =g_k$ if $x\\in F_k$ and $\\widetilde{F}_k \\subset Q_R$.\nSince $\\phi\\in H^1_0(Q_R; \\mathbb{R}^d)$, we have \n\\begin{equation}\\label{p-2}\n\\int_{Q_R\\cap \\omega} A\\nabla u \\cdot \\nabla \\phi\\, dx +\\delta^2 \\int_{Q_R\\cap F} A\\nabla u\\cdot \\nabla \\phi\\, dx=0.\n\\end{equation}\nSince $D \\phi =0$ in $F$, we obtain \n$$\n\\int_{Q_R\\cap \\omega} A\\nabla u \\cdot \\nabla \\phi\\, dx=0.\n$$\nThus, \n\\begin{equation}\\label{3.2-5}\n\\aligned\n\\Big|\n\\int_{Q_R\\cap \\omega} A\\nabla u \\cdot \\nabla (u\\varphi^2) \\, dx\\Big|\n&\\le \\sum_k \\Big|\n\\int_{\\widetilde{F}_k \\setminus \\overline{F}_k}\nA\\nabla u \\cdot \\nabla w_k \\, dx \\Big|\\\\\n& \\le C \\sum_k\n\\| D u\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k) }\n\\| D w_k \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}.\n\\endaligned\n\\end{equation}\nNote that by \\eqref{p-1},\n$$\n\\|\\nabla w_k\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n \\le C \\| u\\varphi^2 -g_k \\|_{H^1(F_k)}\n \\le C \\| D(u \\varphi^2)\\|_{L^2(F_k)},\n $$\n where we have chosen $g_k \\in \\mathcal{R}$ such that the last inequality holds.\n Consequently, \n $$\n \\aligned\n \\|\\nabla w_k\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n&\\le C \\| D u \\|_{L^2(F_k)}\n+ C \\|u \\nabla \\varphi\\|_{L^2(F_k)}\\\\\n& \\le C \\delta^{-2} \\| Du \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n+ C \\|u \\nabla \\varphi\\|_{L^2(F_k)},\n\\endaligned\n$$\nwhere we have used \\eqref{7.1-0} for the last inequality.\nThis, together with \\eqref{3.2-5}, gives\n$$\n\\Big|\n\\int_{Q_R\\cap \\omega} A\\nabla u \\cdot \\nabla (u\\varphi^2) \\, dx\\Big|\n\\le (C \\delta^{-2} +\\varepsilon) \\int_{Q_R} | D u|^2\\, dx\n+ C \\varepsilon^{-1}\\int_{Q_R} |u |^2 | \\nabla \\varphi|^2\\, dx\n$$\nfor any $0< \\varepsilon<1$, where we have used the Cauchy inequality.\nHence,\n$$\n\\int_{Q_{r+3} \\cap \\omega} \n|Du |^2\\, dx\n \\le (C \\delta^{-2} +\\varepsilon) \\int_{Q_R} | D u|^2\\, dx\n+ \\frac{C}{\\varepsilon (R-r)^2}\n\\int_{Q_R} |u|^2 \\, dx,\n$$\nwhich, combined with \\eqref{3.2-1}, yields \\eqref{3.2-0}.\n\\end{proof}\n\nThe following theorem provides a weaker version of the Caccioppoli inequality, that is uniform for $\\delta\\in (1, \\infty)$,\nfor the operator $\\mathcal{L}_\\delta$.\n\n\\begin{thm}\\label{thm-3.3}\nSuppose $1<\\delta< \\infty$.\nLet $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)=0$ in $Q_R$ for some $R\\ge 32$.\nThen, for any $\\ell\\ge 1$,\n\\begin{equation}\\label{3.3-0}\n\\int_{Q_{R\/2}} |\\nabla u|^2\\, dx\n\\le \\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx\n+\\frac{C}{R^{2\\ell} } \\int_{Q_R} |\\nabla u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\ell$, and $\\omega$.\n\\end{thm}\n\n\\begin{proof}\nThe proof uses Lemma \\ref{lemma-3.2} and an iteration argument.\nLet $r_i =R(1-2^{-i})$ for $i=1, 2, \\dots$.\nIt follows from \\eqref{3.2-0} that for $0< \\varepsilon<1$,\n\\begin{equation}\\label{3.3-1}\n\\int_{Q_{r_i}}\n|\\nabla u|^2\\, dx\n\\le \\frac{C}{\\varepsilon (r_{i+1} -r_i)^2}\n\\int_{Q_{r_{i+1}}}|u|^2\\, dx\n+ ( \\varepsilon + C \\delta^{-2}) \\int_{Q_{r_{i+1}}} |\\nabla u|^2\\, dx,\n\\end{equation}\nif $r_{i+1} \\ge 16$ and \n$$\n(1\/2) r_{i+1} \\le r_i \\le r_{i+1} -8.\n$$\nIt is easy to verify that the conditions on $r_i$ are satisfied if $1\\le i \\le k$, where $k$ is the largest integer such that\n$R2^{-k-1} \\ge 8$.\nThus, by an induction argument,\n$$\n\\int_{Q_{r_1}}\n|\\nabla u|^2\\, dx\n\\le \\frac{C_0}{\\varepsilon}\n\\sum_{i=1}^k\n\\frac{ (\\varepsilon + C_0 \\delta^{-2})^{i-1}}{(r_{i+1} -r_i)^2}\n\\int_{Q_R} |u|^2\\, dx\n+ (\\varepsilon + C_0 \\delta^{-2})^k\n\\int_{Q_{r_{k+1}}} |\\nabla u|^2\\, dx,\n$$\nwhere $C_0$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\nSince $r_{i+1}-r_i= 2^{-i-1} R$, we see that \n$$\n\\aligned\n\\int_{Q_{R\/2}} |\\nabla u|^2\\, dx\n & \\le \\frac{4C_0}{\\varepsilon (\\varepsilon + C_0 \\delta^{-2}) R^2 }\n\\sum_{i=1}^k ( 4\\varepsilon + 4 C_0 \\delta^{-2})^i \\int_{Q_R} |u|^2\\, dx\\\\\n& \\qquad\\qquad\n+ (\\varepsilon + C_0 \\delta^{-2})^k \\int_{Q_R} |\\nabla u|^2\\, dx.\n\\endaligned\n$$\nWe now choose $\\varepsilon= 2^{-2\\ell-2} $. It follows that if $4C_0\\delta^{-2}\\le 2^{-2\\ell} $, then\n$$\n\\int_{Q_{R\/2}} |\\nabla u|^2\\, dx\n\\le \\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx\n+ (2^{-2\\ell} )^k \\int_{Q_R} |\\nabla u|^2\\, dx.\n$$\nThis gives \\eqref{3.3-0} for the case $\\delta^2\\ge 2^{2\\ell +2} C_0$, as $2^{k}\\approx R$.\nFinally, we observe that the remaining case $1< \\delta^2 < 2^{2\\ell +2} C_0$ is contained in Theorem \\ref{thm-2.3}.\n\\end{proof}\n\nWe now consider the case $\\delta=\\infty$.\nRecall that $u\\in H^1(\\Omega; \\mathbb{R}^d)$ is called a weak solution of $\\mathcal{L}_\\infty (u)=0$ in $\\Omega$ if\n$Du =0$ in $\\Omega\\cap F$ and \n\\begin{equation}\\label{4.0-0}\n\\int_{\\Omega\\cap \\omega} \nA\\nabla u\\cdot \\nabla v \\, dx =0\n\\end{equation}\nfor any $v \\in H^1_0(\\Omega; \\mathbb{R}^d)$ with $D v =0$ in $\\Omega\\cap F$.\n\n\\begin{thm}\\label{thm-4.1}\nLet $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\infty (u)=0$ in $Q_R$ for some $R\\ge 32$.\nThen, for any $\\ell\\ge 1$,\n\\begin{equation}\\label{4.1-0}\n\\int_{Q_{R\/2} } |\\nabla u|^2\\, dx\n\\le \\frac{C}{R^2} \\int_{Q_R} |u|^2\\, dx\n+ \\frac{C}{R^{2\\ell}} \\int_{Q_R} |\\nabla u|^2\\, dx,\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\ell$, and $\\omega$.\n\\end{thm}\n\n\\begin{proof}\nIn view of the proof of Theorem \\ref{thm-3.3}, it suffices to show\nthat for $(R\/2)\\le r \\le R-8$ and $0<\\varepsilon<1$,\n\\begin{equation}\\label{4.1-1}\n\\int_{Q_r} |\\nabla u|^2\\, dx\n\\le \\frac{C}{\\varepsilon (R-r)^2} \n\\int_{Q_R} |u|^2\\, dx\n+ \\varepsilon \\int_{Q_R} |\\nabla u|^2\\, dx.\n\\end{equation}\nThe proof of \\eqref{4.1-1} is similar to that of Lemma \\ref{lemma-3.2}.\nIndeed, by the second Korn inequality,\n\\begin{equation}\\label{4.1-2}\n\\int_{Q_r} |\\nabla u |^2\\, dx \n\\le C \\int_{Q_{r} \\cap \\omega} |D u |^2\\, dx\n+\\frac{1}{r^2} \\int_{Q_r} |u|^2\\,dx,\n\\end{equation}\nwhere we have used the fact $Du =0$ in $Q_r\\cap F$.\nLet $\\varphi\\in C_0^1(Q_{R-3})$ and $\\phi\\in H^1_0(Q_R; \\mathbb{R}^d)$\nbe the same as in the proof of Lemma \\ref{lemma-3.2}.\nNote that $\\phi|_{F_k} \\in \\mathcal{R}$ for each $F_k$ (if $F_k \\cap Q_{R-3}\n=\\emptyset $, then $\\phi=0$).\nThis allows us to use \\eqref{4.0-0} to obtain \n$$\n\\int_{Q_R\\cap \\omega}\nA\\nabla u\\cdot \\nabla \\phi\\, dx =0.\n$$\nThe rest of the argument is the same as in the proof of Lemma \\ref{lemma-3.2}, without the terms involving \n$C\\delta^{-2}$. We omit the details.\n\\end{proof}\n\n\\begin{remark}\\label{remark-3.1}\nLet $1< \\delta\\le \\infty$\nand $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q_R$\nfor some $R$ sufficiently large.\nIt follows from Theorems \\ref{thm-3.3} and \\ref{thm-4.1} (with $\\ell=1$) that \n\\begin{equation}\\label{re-3.1}\n\\aligned\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2 \\right)^{1\/2}\n & \\le \n C \\left(\\fint_{Q_R} |\\nabla u|^2\\right)^{1\/2} + C \\sup_{s\\le r\\le R}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2}\\\\\n& \\qquad\\qquad\n+ \\frac{C}{s}\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2 \\right)^{1\/2}\n\\endaligned\n\\end{equation}\nfor any $s\\in [16, R]$,\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\nChoose $s$ so large that $Cs^{-1}\\le (1\/2)$.\nThis yields \n\\begin{equation}\\label{re-3.2}\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2 \\right)^{1\/2}\n \\le \n C \\left(\\fint_{Q_R} |\\nabla u|^2\\right)^{1\/2} + C \\sup_{s \\le r\\le R}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2}.\n\\end{equation}\nNote that if $1\\le r< s$, $|Q_r|^{-1\/2} \\| u\\|_{L^2(Q_r)}\n\\le C |Q_s|^{-1\/2} \\| u\\|_{L^2(Q_s)}$.\nAs a result, we obtain \n\\begin{equation}\\label{re-3.3}\n\\sup_{1 \\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2 \\right)^{1\/2}\n \\le \n C \\left(\\fint_{Q_R} |\\nabla u|^2\\right)^{1\/2} + C \\sup_{1\\le r\\le R}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\\end{remark}\n\n\n\n\\section{Large-scale estimates}\\label{section-4}\n\nIn this section we give the proof of Theorem \\ref{main-thm-1}.\nAs we mentioned in Introduction, the approach is based on an idea from \\cite{Armstrong-2020}.\n\nLet $u\\in L^1(Q_{2r})$ for some $r\\in \\mathbb{N}$, define\n\\begin{equation}\\label{5.0-0}\n\\widehat{u} (z) = \\int_{Y+z} u(x)\\, dx,\n\\end{equation}\nwhere $Y=(0, 1)^d$, \nfor any $z\\in \\mathbb{Z}^d $ such that $Y+z\\subset Q_{2r}$.\n\n\n\\begin{lemma}\\label{lemma-5.1}\nLet $u\\in H^1(Q_{2r})$ for some $r\\in \\mathbb{N}$.\nThen\n\\begin{equation}\\label{5.1-0}\n\\left(\\fint_{Q_{2r} } |u|^2\\right)^{1\/2}\n\\le C \\sup_{Y+z\\subset Q_{2r}} |\\widehat{u}(z)|\n+ C \\left(\\fint_{Q_{2r}} |\\nabla u|^2\\right)^{1\/2},\n\\end{equation}\nwhere $C$ depends only on $d$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows by using Poincar\\'e's inequality on each unit cube $Y+z\\subset Q_{2r}$ to obtain \n$$\n\\int_{Y+z} |u|^2\\, dx \\le |\\widehat{u}(z)|^2 + C \\int_{Y+z} |\\nabla u|^2\\, dx\n$$\nand summing the inequality over $z$.\n\\end{proof}\n\nFor a function $f$ defined in $\\mathbb{R}^d$ or $\\mathbb{Z}^d$, let\n\\begin{equation}\\label{5.2}\n\\Delta_j f (x)= f(x+e_j) -f(x)\n\\end{equation}\nfor $1\\le j \\le d$, where $e_j =(0, \\dots, 1, \\dots, 0)$ with $1$ in the $j^{th}$ position.\nFor a multi-index $\\gamma=(\\gamma_1, \\gamma_2, \\dots, \\gamma_d)$,\nwe use the notation $\\Delta^\\gamma f =\\Delta_1^{\\gamma_1} \\Delta_2^{\\gamma_2} \\cdots \\Delta_d^{\\gamma_d} f$.\nLet $\\partial^k f = ( \\Delta^\\gamma f )_{|\\gamma|= k}$ and \n$$\n| \\partial^kf | =\\Big( \\sum_{|\\gamma|=k} |\\Delta^\\gamma f|^2 \\Big)^{1\/2}\n$$\nfor an integer $k\\ge 0$.\nThe following discrete Sobolev inequality will be needed:\n\\begin{equation}\\label{Sob}\n\\sup_{z\\in \\mathbb{Z}^d\\cap \\overline {Q}_{2R}}\n|f(z)|\n\\le C \\sum_{k=0}^N \nR^k \\left(\\frac{1}{R^d}\n\\sum_{z\\in \\mathbb{Z}^d \\cap \\overline{Q}_{4R} }\n|\\partial^k f (z)|^2\\right)^{1\/2},\n\\end{equation}\nwhere $R\\ge 1$ is an integer, $N=[d\/2]+1$, and $C$ depends only on $d$.\nWe refer the reader to \\cite{Stevenson-1991} for a proof of (\\ref{Sob}).\n\n\\begin{lemma}\\label{lemma-5.3}\nLet $u\\in H^1(Q_{4R} )$ for some integer $R\\ge 2$.\nThen, for any integer $r\\in [1, 2R]$,\n\\begin{equation}\\label{5.3-0}\n\\inf_{E\\in \\mathbb{R}}\n\\left(\\fint_{Q_{2r} } |u - E|^2\\right)^{1\/2}\n\\le {Cr}\n\\sum_{k=0}^N\nR^k\n\\left(\\fint_{Q_{4R} } \n|\\nabla \\partial ^k u|^2\\right)^{1\/2}\n+ C \\left(\\fint_{Q_{2r}} |\\nabla u|^2\\right)^{1\/2},\n\\end{equation}\nwhere $N=[d\/2]+1$ and $C$ depends only on $d$.\n\\end{lemma}\n\n\\begin{proof}\nWe may assume $r\\le R-1$; for otherwise,\n\\eqref{5.3-0} (with $N=0$) follows directly from Poincar\\'e's inequality.\nBy \\eqref{5.1-0} we have \n\\begin{equation}\\label{5.3-1}\n\\aligned\n\\left(\\fint_{Q_{2r}} |u-\\widehat{u} (0) |^2\\right)^{1\/2}\n & \\le C \\sup_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{2r}} |\\widehat{u}(z)-\\widehat{u} (0) |\n+ C \\left(\\fint_{Q_{2r}} |\\nabla u|^2\\right)^{1\/2}\\\\\n & \\le C r \\sup_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{2r}} |\\partial \\widehat{u} (z) |\n+ C \\left(\\fint_{Q_{2r}} |\\nabla u|^2\\right)^{1\/2}.\n\\endaligned\n\\end{equation}\nTo bound the first term in the right-hand side of (\\ref{5.3-1}),\nwe use \\eqref{Sob} to obtain \n\\begin{equation}\\label{5.3-2}\n\\sup_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{2R-2} } |\\partial \\widehat{u} (z) |\n\\le \nC \\sum_{k=0}^N\nR^k \\left(\\frac{1}{R^d}\n\\sum_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{4R-4}}\n|\\partial^{k+1} \\widehat{u}(z)|^2 \\right)^{1\/2}.\n\\end{equation}\nNote that \n$$\n\\aligned\n|\\Delta_j \\partial^k \\widehat{u} (z)|^2\n & \\le \\int_{Y+z}\n|\\partial^k u(x+e_j) -\\partial^k u(x)|^2\\, dx\\\\\n&\\le \\int_{Y+z}\n\\int_0^1 |\\nabla \\partial^k u (x+t e_j)|^2 \\, dt\\, dx\\\\\n&\\le \\int_{(0, 2)^d +z} |\\nabla \\partial^k u(x)|^2\\, dx.\n\\endaligned\n$$\nIt follows that\n$$\n\\left(\\frac{1}{R^d}\n\\sum_{z\\in \\mathbb{Z}^d\\cap \\overline{Q}_{4R-4}}\n|\\partial^{k+1} \\widehat{u}(z)|^2 \\right)^{1\/2}\n\\le C \\left(\\fint_{Q_{4R}}\n| \\nabla \\partial ^k u|^2\\, dx\\right)^{1\/2}.\n$$\nThis, together with \\eqref{5.3-1}and \\eqref{5.3-2}, gives \\eqref{5.3-0}.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{main-thm-1}]\n\nLet $u\\in H^1(Q_R; \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u)=0$ in $Q_R$ for some $R\\ge 4$.\nWithout loss of generality we may assume $R$ is a large even integer.\n\nWe first point out that \\eqref{Lip-e-1} follows from \\eqref{Lip-e}.\nIndeed, let $1\\le r\\le R-3$. Note that for any $\\phi \\in \\mathcal{R}$, we have $\\mathcal{L}_\\delta (u-\\phi)=0$ in $Q_R$. \nIt follows that\n$$\n\\aligned\n\\left(\\fint_{Q_r} |D u|^2 \\right)^{1\/2}\n &\\le C \\left(\\fint_{Q_{R-3}} |\\nabla (u-\\phi)|^2\\right)^{1\/2}\\\\\n &\\le C \\left(\\fint_{Q_{R-3}} |Du|^2 \\right)^{1\/2},\n \\endaligned \n $$\n where we have chosen $\\phi\\in \\mathcal{R}$ so that the last inequality holds.\n This, together with \\eqref{ex-1a}, gives \\eqref{Lip-e-1}.\n The rest of the proof is devoted to (\\ref{Lip-e}.\n In view of \\eqref{ex-1a} it suffices to bound the left-hand side of \\eqref{Lip-e})\n by the $L^2$ average of $|\\nabla u|$ over $Q_R$.\n \n\\medskip\n\n\\noindent{\\bf Case I: $0 \\le \\delta\\le 1$.}\nNote that $\\mathcal{L}_\\delta (\\Delta^\\gamma u )=0$ in $Q_{R-2k}$\nfor any multi-index $\\gamma$ with $|\\gamma|=k$.\nIt follows from Theorem \\ref{thm-2.3} that\n$$\n\\fint_{Q_{\\rho }} |\\nabla \\partial^k u |^2\n\\le \\frac{C}{\\rho ^2} \\fint_{Q_{2\\rho}}\n| \\partial^k u|^2\\, dx\n$$\nfor $4\\le \\rho< (R-2k)\/2$, where we have used the condition $0\\le\\delta\\le 1$.\n This, together with the observation that \n\\begin{equation}\\label{5.4-1}\n\\fint_{Q_{2\\rho} } |\\partial^k u|^2\\, dx\n\\le C \\fint_{Q_{2\\rho+2}} |\\nabla \\partial^{k-1} u|^2\\, dx\n\\end{equation}\nfor $k \\ge 1$, yields\n\\begin{equation}\\label{5.4-2}\n\\fint_{Q_{\\rho }} |\\nabla \\partial^k u |^2\n\\le \\frac{C}{\\rho^2} \\fint_{Q_{2\\rho+2}} |\\nabla \\partial^{k-1} u|^2\\, dx.\n\\end{equation}\nBy induction we obtain\n\\begin{equation}\\label{5.4-3}\n\\fint_{Q_{cR}}\n|\\nabla \\partial^k u|^2\\, dx\n\\le \\frac{C}{\\rho^{2k}}\n\\fint_{Q_R} |\\nabla u|^2,\n\\end{equation}\nwhere $C$ and $c$ depend only on $d$, $k$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\nBy combining \\eqref{5.4-3} with \\eqref{5.3-0} we see that\nfor any $r\\in [1, cR]$,\n\\begin{equation}\\label{5.4-4}\n\\inf_{E\\in \\mathbb{R}^d}\n\\frac{1}{r}\n\\left(\\fint_{Q_r} |u-E|^2 \\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |\\nabla u|^2 \\right)^{1\/2}\n+\\frac{C}{r}\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}.\n\\end{equation}\nBy Poincar\\'e's inequality we see that the inequality above also holds for $r\\in [cR, R]$.\nHence, if $11$ so large that $C s^{-1}\\le (1\/2)$.\nThis leads to \n$$\n\\sup_{s\\le r\\le R}\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |\\nabla u|^2 \\right)^{1\/2}.\n$$\nThe estimate for the case $1\\le r< s$ follows from the case $r=s$.\n\n\\medskip\n\n\\noindent{\\bf Case II: $1<\\delta\\le \\infty$.}\nAs in the case $0\\le \\delta\\le 1$, $\\mathcal{L}_\\delta (\\Delta^\\gamma u)=0$ in $Q_{R-2k}$ for any multi-index $\\gamma$\nwith $|\\gamma|=k$.\nIt follows from Theorems \\ref{thm-3.3} and \\ref{thm-4.1} (with $\\ell=1$) that\n\\begin{equation}\\label{5.4-10}\n\\fint_{Q_{\\rho}}\n|\\nabla \\partial^k u|^2\n\\le \\frac{C}{\\rho^2}\n\\fint_{Q_{2\\rho} } | \\partial^k u|^2 +\\frac{C}{\\rho^2}\n\\fint_{Q_{2\\rho} } |\\nabla \\partial^k u |^2\n\\end{equation}\nif $16 \\le \\rho\\le (R-2k)\/2$.\nThis, together with \\eqref{5.4-1} and a simple observation that\n$$\n\\fint_{Q_{2\\rho}}\n|\\nabla \\partial^k u|^2\n\\le C \\fint_{Q_{2\\rho+2}}\n|\\nabla \\partial^{k-1} u|^2,\n$$\ngives (\\ref{5.4-2}).\nAs a result, the inequality (\\ref{5.4-4-0})\ncontinues to hold for the case $1<\\delta\\le \\infty$.\nIn view of Remark \\ref{remark-3.1},\nthe inequality (\\ref{5.4-4-00}) also holds for $1<\\delta\\le \\infty$. \nThe rest of the proof is the same as in Case I.\n\\end{proof}\n\nIt follows from Theorem \\ref{main-thm-1} and Poincar\\'e's inequality that if $\\mathcal{L}_\\delta (u)=0$ in $Q_R$ for some \n$R\\ge 1$, then\n\\begin{equation}\\label{L-inf}\n\\sup_{1\\le r\\le R} \\left(\\fint_{Q_r} |u|^2\\right)^{1\/2}\n\\le C \\left(\\fint_{Q_R} |u|^2 \\right)^{1\/2}\n+ C R^2 \\left(\\fint_{Q_R} |\\nabla u|^2\\right)^{1\/2},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n\n\n\n\\section{Local Lipschitz estimates}\\label{section-5}\n\nThroughout this section we will assume $A$ satisfies the elasticity condition (\\ref{ellipticity}) and\nH\\\"older continuity condition (\\ref{smoothness}).\nThe periodicity condition is not needed.\nFor $0 \\psi (x^\\prime) \\big\\},\\\\\nQ_r^- & =\\big\\{ x=(x^\\prime, x_d) \\in Q_r: x_d< \\psi (x^\\prime) \\big\\},\\\\\nI_r & =\\big\\{ x=(x^\\prime, x_d) \\in Q_r: x_d= \\psi (x^\\prime) \\big\\},\\\\\n\\endaligned\n\\end{equation}\nwhere $\\psi: \\mathbb{R}^{d-1} \\to \\mathbb{R}$ is a $C^{1, \\sigma}$ function for some $\\sigma \\in (0, 1)$\nsuch that $\\psi(0)=0$ and $\\|\\nabla \\psi \\|_{C^{1, \\sigma}(\\mathbb{R}^{d-1})} \\le M_1$.\nLet $0<\\delta<\\infty$ and $u\\in H^1(Q_r; \\mathbb{R}^d)$ be a solution of \n\\begin{equation}\\label{local-1}\n\\left\\{\n\\aligned\n -\\text{\\rm div} (A\\nabla u) & = \\delta^{-2} f &\\quad & \\text{ in } Q_r^+,\\\\\n -\\text{\\rm div} (A\\nabla u) & =f & \\quad & \\text{ in } Q_r^-,\\\\\n \\delta^2 \\frac{\\partial u}{\\partial \\nu_+} & = \\frac{\\partial u}{\\partial \\nu_-} &\\quad & \\text{ on } I_r,\n\\endaligned\n\\right.\n\\end{equation}\nwhere $\\frac{\\partial u}{\\partial \\nu_\\pm}\n=n\\cdot A(\\nabla u)_\\pm$ and $\\pm$ indicates the limit taken from $Q_r^\\pm$, respectively.\nIf $\\delta=0$, by a solution of \\eqref{local-1}, we mean that $-\\text{\\rm div} (A\\nabla u)=f$ in $Q_r^-$,\n$-\\text{\\rm div} (A\\nabla u)=0$ in $Q_r^+$, and that $\\frac{\\partial u}{\\partial \\nu_-} =0$ on $I_r$.\nIf $\\delta=\\infty$, the equation \\eqref{local-1} is understood in the sense that $u|_{Q_r^+} \\in \\mathcal{R}$ and \n$-\\text{\\rm div}(A\\nabla u)=f$ in $Q_r^-$.\n\n\\begin{lemma}\\label{local-lemma-1}\nAssume that $A$ satisfies \\eqref{ellipticity} and \\eqref{smoothness}.\nLet $0\\le \\delta\\le 1$ and $u\\in H^1(Q_r; \\mathbb{R}^d)$ be a weak solution of \\eqref{local-1} for some $0d$ and $C$ depends only on $d$, $ p$, $\\kappa_1$, $\\kappa_2$, $\\sigma$, $M_0$, and $M_1$.\n\\end{lemma}\n\n\\begin{proof}\nThe case $0< \\delta_0\\le \\delta\\le 1$ with a constant $C$ depending on $\\delta_0$ \nfollows from the classical results on Lipschitz estimates for elliptic systems \nwith piecewise H\\\"older continuous coefficients.\nIndeed, since $\\psi$ is $C^{1, \\sigma}$, the problem may be reduced to the case\n$\\psi=0$ by flatting the boundary.\nOne may further reduce the problem to the case of constant coefficients by a Campanato perturbation argument.\nWe refer the reader to \\cite{Li-2000, Dong-2012} and their references for more recent development.\n\nWe now treat the case where $\\delta$ is small.\nBy rescaling we may assume $r=1$.\nLet $0<\\rho<1$ and $0<\\tau < \\sigma$. \nSince $\\text{\\rm div} (A\\nabla u)=f$ in $Q_1^-$,\nby the classical $C^{1, \\tau} $ estimates for Neumann problems,\n\\begin{equation}\\label{local-3}\n\\| \\nabla u\\|_{C^{\\tau}(Q^-_{\\rho\/2})}\n\\le C\\Big \\|\\frac{\\partial u}{\\partial \\nu_-}\\Big \\|_{C^{\\tau} (I_{\\rho})}\n+\\frac{C}{\\rho^{1+\\tau} }\n\\left\\{\n\\left(\\fint_{Q_\\rho^-} |u|^2\\right)^{1\/2}\n+ \\rho^2 \\left(\\fint_{Q_\\rho^-} |f|^p \\right)^{1\/p} \\right\\}.\n\\end{equation}\nLet $1\/2\\le s0$,\na similar argument using $C^{1, \\tau}$ estimates for the Dirichlet problem gives\n\\begin{equation}\\label{local-5}\n \\| \\nabla u\\|_{C^\\tau (Q_{t_1}^+)}\n\\le C \\|\\nabla u\\|_{C^\\tau (Q_t^-)}\n+ C (t-s)^{-\\frac{d}{2}-1-\\tau }\n\\left\\{\n \\left(\\fint_{Q_1^+} |u|^2\\right)^{1\/2}\n +\\delta^{-2} \\left( \\fint_{Q_1^+} |f|^p \\right)^{1\/p} \\right\\}.\n\\end{equation}\nBy combining (\\ref{local-4}) with (\\ref{local-5}) it follows that \n\\begin{equation}\\label{local-6}\n\\aligned\n\\| \\nabla u\\|_{C^{\\tau} (Q_s^-)}\n & \\le C \\delta^2 \\|\\nabla u\\|_{C^\\tau (Q_t^-)}\n+ C (t-s)^{-\\frac{d}{2}-1-\\tau }\n\\left\\{\n \\left(\\fint_{Q_1^-} |u|^2\\right)^{1\/2}\n + \\delta^2\n\\left(\\fint_{Q_1^+} |u|^2\\right)^{1\/2} \\right\\}.\\\\\n&\\qquad\\qquad\n+ C (t-s)^{-\\frac{d}{2} -1 -\\tau} \n\\left(\\fint_{Q_1} |f|^p \\right)^{1\/p},\n\\endaligned\n\\end{equation}\nwhich also holds for the case $\\delta=0$.\n Let $s_i=1-2^{-i}$.\n By taking $s=s_i$ and $t=s_{i+1}$ in \\eqref{local-6} and using iteration,\n we see that\n \\begin{equation}\\label{local-7}\n \\aligned\n & \\| \\nabla u\\|_{C^\\tau (Q_{1\/2}^-)}\\\\\n & \\le \n C \\sum_{i=1}^N (s_{i+1}-s_i)^{-\\frac{d}{2} -1-\\tau} \\delta^{2 (i-1)}\n \\left\\{\n \\left(\\fint_{Q_1^-} |u|^2\\right)^{1\/2}\n + \\delta^2\n\\left(\\fint_{Q_1^+} |u|^2\\right)^{1\/2}\n+ \\left(\\fint_{Q_1} |f|^p \\right)^{1\/p} \\right\\}\\\\\n& \\qquad\\qquad\n+ C \\delta^{2N} \\| \\nabla u\\|_{C^{\\tau} (Q_{s_{N+1}})}\n\\endaligned\n\\end{equation}\nfor any $N\\ge 1$.\nNote that $s_{i+1}-s_i= 2^{-i-1}$.\nIt follows that if $2^{\\frac{d}{2} +1 +\\tau} \\delta^2 \\le (1\/2)$, we may let $N \\to \\infty$ in \\eqref{local-7}\nto obtain the first inequality in \\eqref{local-2} with $r=1$.\nThe second inequality follows from the the first and \\eqref{local-5}.\n\\end{proof}\n\n\\begin{remark}\\label{remark-5.1a}\nWe may replace (\\ref{local-2}) by \n\\begin{equation}\\label{local-2b}\n\\left\\{\n\\aligned\n\\|\\nabla u\\|_{L^\\infty (Q_{r\/2}^-) }\n&\\le C \n\\left\\{ \n \\delta^2 \\left(\\fint_{Q_r^+} |\\nabla u|^2 \\right)^{1\/2}\n+ \\left(\\fint_{Q_r^-} |\\nabla u|^2 \\right)^{1\/2}\n+r \\left( \\fint_{Q_r} |f|^p \\right)^{1\/p}\\right\\},\\\\\n\\| \\nabla u\\|_{L^\\infty (Q_{r\/2}^+ ) }\n & \\le C \n\\left\\{\n\\left(\\fint_{Q_r} |\\nabla u|^2\\right)^{1\/2}\n+ r \\left(\\fint_{Q^-_r} |f|^p \\right)^{1\/p} \n+ \\delta^{-2} r\n\\left(\\fint_{Q_r^+} |f|^p \\right)^{1\/p} \\right\\},\n\\endaligned\n\\right.\n\\end{equation}\nTo see this, in the proof of Lemma \\ref{local-lemma-1},\none replaces $u$ in \\eqref{local-3} and (\\ref{local-4}) by $u-E$ and applies \nPoincar\\'e's inequality.\n\\end{remark}\n\n\\begin{remark}\\label{remark-5.1}\nUnder the same assumptions as in Lemma \\ref{local-lemma-1}, we have \n\\begin{equation}\\label{local-2-r}\n\\left\\{\n\\aligned\n\\| u\\|_{L^\\infty (Q_{r\/2}^-) }\n&\\le C\n\\left\\{ \n \\delta^2 \\left(\\fint_{Q_r^+} |u|^2 \\right)^{1\/2}\n+ \\left(\\fint_{Q_r^-} |u|^2 \\right)^{1\/2}\n+r^2 \\left( \\fint_{Q_r} |f|^p \\right)^{1\/p}\\right\\},\\\\\n\\| u\\|_{L^\\infty (Q_{r\/2}^+ ) }\n & \\le C \n\\left\\{\n\\left(\\fint_{Q_r} |u|^2\\right)^{1\/2}\n+ r^2 \\left(\\fint_{Q^-_r} |f|^p \\right)^{1\/p} \n+ \\delta^{-2} r^2 \n\\left(\\fint_{Q_r^+} |f|^p \\right)^{1\/p} \\right\\},\n\\endaligned\n\\right.\n\\end{equation}\nThis follows readily from \\eqref{local-2} and the Mean Value Theorem.\n\\end{remark}\n\nThe next lemma treats the case $1<\\delta\\le \\infty$.\n\n\\begin{lemma}\\label{local-lemma-2}\nAssume that $A$ satisfies (\\ref{ellipticity} and (\\ref{smoothness}).\nLet $1<\\delta\\le \\infty$ and $u\\in H^1(Q_r; \\mathbb{R}^d)$ be a weak solution of \\eqref{local-1} for some $0d$.\nThen \n\\begin{equation}\\label{5.5-0}\n\\aligned\n & |u(x_0)| + r|\\nabla u (x_0)|\\le \\\\\n &\n \\left\\{\n \\aligned\n & C \\left(\\fint_{Q(x_0, r) } |\\Lambda_{\\delta^2} u|^2 \\right)^{1\/2} \n + Cr^2 \\left(\\fint_{Q(x_0, r)} | f|^p \\right)^{1\/p} & & \\text{ if } 0\\le \\delta \\le 1 \\text{ and } x_0 \\in \\omega,\\\\\n& C \\left(\\fint_{Q(x_0, r) } |u|^2 \\right)^{1\/2} \n+ Cr^2 \\left(\\fint_{ Q(x_0, r)}\n|\\Lambda_{\\delta^{-2}} f|^p \\right)^{1\/P} & & \\text{ if } 0\\le \\delta \\le 1 \\text{ and } x_0 \\in F,\n\\endaligned\n\\right.\n\\endaligned\n\\end{equation}\nand\n\\begin{equation}\\label{5.5-1}\n\\aligned\n& |u(x_0)| + r|\\nabla u (x_0)|\\le \\\\\n & \n \\left\\{\n \\aligned\n & C \\left(\\fint_{Q(x_0, r) } | u|^2 \\right)^{1\/2}\n + C r^2\n \\left(\\fint_{Q(x_0, r)} | \\Lambda_{\\delta^{-2}} f|^p \\right)^{1\/p} & & \\text{ if } 1< \\delta < \\infty \\text{ and } x_0 \\in \\omega,\\\\\n& \\frac{ C}{\\delta^2 } \\left(\\fint_{Q(x_0, r) } |\\Lambda_{\\delta^2} u|^2 \\right)^{1\/2} \n+ C\\delta^{-2} r^2\n\\left(\\fint_{Q_r} |f|^p \\right)^{1\/p} & & \\text{ if } 1< \\delta < \\infty \\text{ and } x_0 \\in F.\n\\endaligned\n\\right.\n\\endaligned\n\\end{equation}\nIf $\\delta=\\infty$, we have\n\\begin{equation}\\label{5.5-2}\n\\aligned\n|u(x_0)| + r|\\nabla u (x_0)|\n & \\le C \\left(\\fint_{Q(x_0, r) } | u|^2 \\right)^{1\/2} \n + C r^2 \\left(\\fint_{Q(x_0, r)\\cap \\omega} |f|^p \\right)^{1\/p}& \\quad & \\text{ if } x_0 \\in \\omega\n \\\\\n|u(x_0)| + r|\\nabla u (x_0)|\n& \\le C \\left(\\fint_{Q(x_0, r)\\cap F } | u|^2 \\right)^{1\/2} & \\quad & \\text{ if } x_0 \\in F.\n\\endaligned\n\\end{equation}\nThe constant $C$ depends only on $d$, $p$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$ in \\eqref{smoothness}.\n\\end{thm}\n\n\\begin{proof}\nNote that $-\\text{\\rm div}(A\\nabla u)=f$ in $Q(x_0, r)\\cap \\omega$ and \n$-\\text{\\rm div}(A\\nabla u)=\\delta^{-2}f$ in $Q(x_0, r)\\cap F$.\nIf $Q(x_0, cr)\\subset \\omega$ or $F$ for some small $c>0$, the estimates in \\eqref{5.5-0}-\\eqref{5.5-2} follow directly from the interior estimates \nfor solutions of $-\\text{\\rm div}(A\\nabla u)=f $. \nIn the case $Q(x_0, cr)\\cap \\partial \\omega \\neq \\emptyset$,\none may find $y_0\\in \\partial \\omega$ such that $x_0\\in Q(y_0, r\/2)$ and $Q(y_0, r\/2)\\subset Q(x_0, r)$.\nAs a result, the estimates in \\eqref{5.5-0}-\\eqref{5.5-2} follow readily from \\eqref{local-2}, \\eqref{local-2-r},\n\\eqref{local-2-0} and \\eqref{local-3-0} by a simple localization argument.\n\\end{proof}\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{main-thm-2}]\n\nNote that for all cases in Theorem \\ref{local-thm-0} with $f=0$,\n\\begin{equation}\\label{5.6-0}\n|u(x_0)| + r|\\nabla u(x_0)| \\le C \\left(\\fint_{Q(x_0, r)} |u|^2\\right)^{12}.\n\\end{equation}\nSince $u-E$ is also a solution for any $E\\in \\mathbb{R}^d$, one may use Poincar\\'e's inequality to obtain \n\\begin{equation}\\label{5.6-1}\n|\\nabla u(x_0)| \\le C \\left(\\fint_{Q(x_0, r)} |\\nabla u|^2\\right)^{12}\n\\end{equation}\nfor $00$, then \n$$\n|\\nabla u(x_0)|\\le C_\\delta \\left(\\fint_{Q(x_0, R)} |\\nabla u|^2 \\right)^{1\/2},\n$$\nwhere $C_\\delta$ may depend on $\\delta$. It follows that the operator $\\mathcal{L}_\\delta$ possesses \na fundamental solution $\\Gamma_\\delta (x, y) = \\big( \\Gamma_\\delta^{\\alpha \\beta} (x, y) \\big)_{d\\times d}$ \nin the sense that if \n\\begin{equation}\\label{f-1}\nu(x)=\\int_{\\mathbb{R}^d} \\Gamma_\\delta (x, y) f (y)\\, dy\n\\end{equation}\nfor some $f\\in C_0^\\infty (\\mathbb{R}^d; \\mathbb{R}^d)$, then $u\\in L^{2^*} (\\mathbb{R}^d; \\mathbb{R}^d)$, $\\nabla u\\in L^2(\\mathbb{R}^d; \\mathbb{R}^{d\\times d})$, and \n\\begin{equation}\\label{f-2}\n\\int_{\\mathbb{R}^d} \\Lambda_{\\delta^2} A\\nabla u \\cdot \\nabla v \\, dx\n=\\int_{\\mathbb{R}^d} f \\cdot v \\, dx\n\\end{equation}\nfor any $v \\in L^{2^*} (\\mathbb{R}^d; \\mathbb{R}^d)$ with $\\nabla v \\in L^2 (\\mathbb{R}^d; \\mathbb{R}^d)$, where $2^*=\\frac{2d}{d-2}$.\nMoreover, there exists a constant $C_\\delta$ such that\n$|\\Gamma_\\delta (x, y)|\\le C_\\delta |x-y|^{2-d}$ and\n$|\\nabla_x \\Gamma_\\delta (x, y)| + |\\nabla_y \\Gamma_\\delta (x, y)|\\le C_\\delta |x-y|^{1-d}$\nfor any $x, y\\in \\mathbb{R}^d$.\nWe refer the reader to \\cite{Hofmann-2007} for the construction of $\\Gamma_\\delta(x, y)$\nunder a H\\\"older continuity condition on weak solutions.\nOur goal of this section is to establish the explicit dependence of $C_\\delta$ on $\\delta$.\n\n\\begin{lemma}\\label{lemma-f-1}\nLet $u$ be given by \\eqref{f-1} with $f\\in C_0^\\infty(\\omega; \\mathbb{R}^d)$.\nThen\n\\begin{equation}\\label{f-1-0}\n\\delta \\| Du \\|_{L^2( F)} \n+ \\|\\nabla u \\|_{L^2(\\omega)} +\\| u\\|_{L^{p^\\prime} (\\omega)} \n \\le C \\| f\\|_{L^p(\\omega)},\n\\end{equation}\nwhere $p=\\frac{2d}{d+2}$ and $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n \n \\begin{proof}\n By letting $v=u$ in \\eqref{f-2} we obtain \n \\begin{equation}\\label{f-1-1}\n \\delta^2 \\int_{F} |D u |^2\\, dx \n +\\int_\\omega |Du |^2\\, dx \\le C \\| f\\|_{L^p(\\omega)} \\| u\\|_{L^{p^\\prime } (\\omega)},\n \\end{equation}\n where $p=\\frac{2d}{d+2}$. \n Next, let $U$ be an extension of $u$ from $\\omega$ to $\\mathbb{R}^d$ such that\n \\begin{equation}\\label{f-1-3}\n \\| \\nabla U\\|_{L^2(\\mathbb{R}^d)} \\le C \\| \\nabla u\\|_{L^2(\\omega)} \\quad \\text{ and } \\quad\n \\| D U \\|_{L^2(\\mathbb{R}^d)} \\le C \\| D u \\|_{L^2(\\omega)} .\n \\end{equation}\n The function $U$ may be obtained by extending $u$ from $\\widetilde{F}_k \\setminus F_k$ to $F_k$, for each $k$,\n so that \n \\begin{equation}\\label{f-1-3a}\n \\| \\nabla U\\|_{L^2(\\widetilde{F}_k) } \\le C \\| \\nabla u\\|_{L^2(\\widetilde{F} _k \\setminus \\overline{F}_k)}\n \\quad \\text{ and } \\quad\n \\| D U \\|_{L^2(\\widetilde{F}_k) } \\le C \\| D u \\|_{L^2(\\widetilde{F} _k \\setminus \\overline{F}_k)}.\n \\end{equation}\n See Lemma \\ref{lemma-ex}.\n Since $|u(x)| +|x| |\\nabla u (x)| =O(|x|^{2-d})$ as $|x|\\to \\infty$ and $d\\ge 3$, we see that\n \\begin{equation}\\label{decay}\n \\frac{1}{R^2} \\int_{Q_{2R} \\setminus Q_R} \\big ( |u|^2 +|\\nabla u|^2 \\big) \\, dx \\to 0 \\quad \\text{ as } R \\to \\infty.\n \\end{equation}\n The property (\\ref{ex-2}) also implies that $U$ satisfies the condition (\\ref{decay}).\n This allows us to apply the first Korn inequality and Sobolev inequality in $Q_R$ and then let $R\\to \\infty$ to deduce that\n \\begin{equation}\\label{f-1-1a}\n \\| \\nabla U\\|_{L^2(\\mathbb{R}^d)} \\le C \\| D U \\|_{L^2(\\mathbb{R}^d)}\n \\quad\n \\text{ and }\n \\quad\n \\| U\\|_{L^{p^\\prime}(\\mathbb{R}^d)} \\le C \\| \\nabla U \\|_{L^2(\\mathbb{R}^d)}.\n\\end{equation}\n As a result, we obtain \n \\begin{equation}\\label{f-1-4}\n \\| U \\|_{L^{p^\\prime}(\\mathbb{R}^d)}\n \\le C \\| \\nabla U \\|_{L^2(\\mathbb{R}^d)}\n \\le C \\| D U \\|_{L^2(\\mathbb{R}^d)}\n \\le C \\| D u \\|_{L^2(\\omega)}.\n \\end{equation}\n It follows that\n \\begin{equation}\\label{f-1-4a}\n \\aligned\n & \\| \\nabla u\\|_{L^2(\\omega)} =\\| \\nabla U \\|_{L^2(\\omega)} \\le C \\| D u \\|_{L^2(\\omega)},\\\\\n & \\| u\\|_{L^{p^\\prime}(\\omega)}= \\| U \\|_{L^{p^\\prime}(\\omega)}\n \\le C \\| D u \\|_{L^2(\\omega)}.\n \\endaligned\n \\end{equation}\n Consequently, by \\eqref{f-1-1} and the Cauchy inequality, \n we see that $ \\delta \\| D u \\|_{L^2(F)} \\le C \\| f\\|_{L^p(\\omega)}$ and\n $$\n \\|\\nabla u\\|_{L^2(\\omega)}\n + \\| u\\|_{L^{p^\\prime} (\\omega)} \n \\le C \\| Du \\|_{L^2(\\omega)} \n \\le C \\| f\\|_{L^p(\\omega)}, \n $$\n which completes the proof.\n \\end{proof}\n \n \\begin{remark}\\label{re-div}\n Let $u$ be given by (\\ref{f-1}) with $f=\\text{\\rm div} (g)$,\n where $g\\in C_0^\\infty (\\omega; \\mathbb{R}^{d\\times d})$.\n Then\n $$\n \\delta^2 \\int_{F} | Du |^2\\, dx \n + \\int_{\\omega} |Du |^2\\, dx \\le C \\| g \\|_{L^2(\\omega)} \\| \\nabla u\\|_{L^2(\\omega)}.\n $$\n Using (\\ref{f-1-4a}), we obtain \n \\begin{equation}\\label{re-div-0}\n \\delta\n \\| Du \\|_{L^2(F)} + \\| \\nabla u\\|_{L^2(\\omega)}\n + \\| u\\|_{L^{p^\\prime} (\\omega)}\n \\le C \\| g\\|_{L^2(\\omega)},\n \\end{equation}\n where $p^\\prime=\\frac{2d}{d-2}$ and $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{remark}\n \n \\begin{remark}\\label{re-large}\n \n Suppose $1\\le \\delta< \\infty$.\n Let $u$ be given by \\eqref{f-1} with $f\\in C_0^\\infty (\\mathbb{R}^d; \\mathbb{R}^d)$.\n By letting $v=u$ in \\eqref{f-2} we obtain \n $\\| D u \\|^2_{L^2(\\mathbb{R}^d)} \\le C \\| f\\|_{L^p(\\mathbb{R}^d)} \\| u\\|_{L^{p^\\prime}(\\mathbb{R}^d)}$.\n Using \n $$\\| u\\|_{L^{p^\\prime}(\\mathbb{R}^d)} \\le C \\| \\nabla u\\|_{L^2(\\mathbb{R}^d)} \\le C \\| D u \\|_{L^2 (\\mathbb{R}^d)},\n $$\n we see that $\\|u\\|_{L^{p^\\prime}(\\mathbb{R}^d)} \\le C \\| f\\|_{L^p(\\mathbb{R}^d)}$.\n \n \\end{remark}\n \n \\begin{thm}\\label{thm-f-1}\n Let $0<\\delta<\\infty$.\n For $x, y\\in \\mathbb{R}^d$ with $|x-y|_\\infty\\ge 4$, we have\n \\begin{align}\n |\\Gamma_\\delta (x, y)| & \\le C |x-y|^{2-d} , \\label{f-2-0}\\\\\n |\\nabla_x \\Gamma_\\delta (x, y)|\n +|\\nabla _y \\Gamma_\\delta (x, y)|\n & \\le C |x-y|^{1-d} , \\label{f-2-1} \\\\\n |\\nabla_x\\nabla_y \\Gamma_\\delta (x, y)| & \\le C |x-y|^{-d}, \\label{f-2-00}\n \\end{align}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$.\n \\end{thm}\n \n\\begin{proof}\nFix $x_0, y_0\\in \\mathbb{R}^d$ with $r=|x_0-y_0|_\\infty\\ge 4$.\nLet $u$ be given by \\eqref{f-1} with $f\\in C_0^\\infty (\\omega\\cap Q(y_0, r); \\mathbb{R}^d) $.\nSince $\\mathcal{L}_\\delta (u) =0$ in $Q (x_0, r)$,\nit follows from (\\ref{5.6-0}) and \\eqref{L-inf} that\n\\begin{equation}\\label{f-2-2}\n\\aligned\n|u(x_0)|\n& \\le C \\left(\\fint_{Q(x_0, 1\/2)} |u|^2\\right)^{1\/2}\\\\\n& \\le C \\left(\\fint_{Q(x_0, r\/4)}\n|u|^2 \\right)^{1\/2}\n+ C r \\left(\\fint_{Q(x_0, r\/4)} |\\nabla u|^2 \\right)^{1\/2}\\\\\n& \\le C \\left(\\fint_{Q(x_0, 3+r\/4)\\cap \\omega}\n|u|^2 \\right)^{1\/2}\n+ C r \\left(\\fint_{Q(x_0, 3+ r\/4) \\cap \\omega} |\\nabla u|^2 \\right)^{1\/2},\n\\endaligned\n\\end{equation}\nwhere we have used \\eqref{ex-1a} and \\eqref{ex-3a} for the last inequality.\nWe now use (\\ref{f-1-0}) to bound the right-hand side of \\eqref{f-2-2}.\nThis gives\n$$\n\\aligned\n| u(x_0)|\n&\\le C r^{1-\\frac{d}{2}} \\big\\{ \\| u\\|_{L^{p^\\prime}(\\omega)}\n+ \\| \\nabla u\\|_{L^2(\\omega)} \\big\\}\\\\\n&\\le C r^{1-\\frac{d}{2}} \\| f\\|_{L^p(\\omega)},\n\\endaligned\n$$\nwhere $p=\\frac{2d}{d+2}$.\nBy duality it follows that \n\\begin{equation}\\label{f-2-3}\n\\left(\\int_{\\omega\\cap Q(y_0, r)}\n| \\Gamma_\\delta (x_0, y)|^{p^\\prime} \\, dy \\right)^{1\/p^\\prime}\n\\le C r^{1-\\frac{d}{2}}.\n\\end{equation}\nNote that if $f=\\text{\\rm div} (g)$, where $g\\in C_0^\\infty(\\omega\\cap Q(y_0, r); \\mathbb{R}^{d\\times d} )$,\nwe may use \\eqref{f-2-2} and \\eqref{re-div-0} to obtain \n$$\n|u(x_0)|\\le C r^{1-\\frac{d}{2}} \\| g\\|_{L^2(\\omega)}.\n$$\nBy duality we deduce that\n\\begin{equation}\\label{f-2-4}\n\\left(\\int_{\\omega\\cap Q(y_0, r)}\n| \\nabla_y \\Gamma_\\delta (x_0, y)|^{2} \\, dy \\right)^{1\/2}\n\\le C r^{1-\\frac{d}{2}}.\n\\end{equation}\nAlso, note that by Theorem \\ref{main-thm-2},\n$$\n|\\nabla u(x_0)|\\le C \\left(\\fint_{\\omega\\cap Q(x_0, r)} |\\nabla u|^2 \\right)^{1\/2}\n\\le Cr^{-\\frac{d}{2}} \\| g \\|_{L^2(\\omega)}.\n$$\nAgain, by duality, we obtain \n\\begin{equation}\\label{dd}\n\\left(\\int_{\\omega\\cap Q(y_0, r)}\n| \\nabla_x \\nabla_y \\Gamma_\\delta (x_0, y)|^{2} \\, dy \\right)^{1\/2}\n\\le C r^{-\\frac{d}{2}}.\n\\end{equation}\n\nNow, let $ v(y)= \\Gamma_\\delta (x_0, y)$.\nThen $\\mathcal{L}_\\delta^* (v)=0$ in $Q(y_0, r)$, where $\\mathcal{L}_\\delta^*$ denotes the adjoint of\n$\\mathcal{L}_\\delta$.\nSince $\\mathcal{L}_\\delta^*$ satisfies the same conditions as $\\mathcal{L}_\\delta$,\nwe may use \\eqref{f-2-2} to obtain\n$$\n\\aligned\n|v(y_0)|\n& \n\\le C \\left(\\fint_{\\omega\\cap Q(y_0, 3+r\/4)}\n|v|^2 \\right)^{1\/2}\n+ C r \\left(\\fint_{\\omega\\cap Q(y_0, 3+ r\/4) } |\\nabla v|^2 \\right)^{1\/2}\\\\\n& \\le C r^{2-d},\n\\endaligned\n$$\nwhich gives \\eqref{f-2-0}. Also, note that by Theorem \\ref{main-thm-2},\n\\begin{equation}\\label{f-2-5}\n|\\nabla v (y_0)|\n \\le C \\left(\\fint_{\\omega\\cap Q(y_0, r) } |\\nabla v|^2 \\right)^{1\/2}.\n\\end{equation}\nThis, together with \\eqref{f-2-4}, gives \n$|\\nabla_y \\Gamma_\\delta (x_0, y_0)|\\le C r^{1-d}$.\nThe estimate $|\\nabla_x \\Gamma_\\delta (x_0, y_0)|\\le C r^{1-d}$ \nfollows from the fact that the fundamental solution $\\Gamma^*_\\delta (x, y)$ for $\\mathcal{L}_\\delta^*$ is given by the\ntranspose of $\\Gamma_\\delta (y, x)$.\nFinally, the estimate for $\\nabla_x \\nabla_y \\Gamma_\\delta (x_0, y_0)$\nfollows from \\eqref{dd} and the fact that\n$\\mathcal{L}_\\delta^* (\\nabla_x \\Gamma_\\delta (x_0, \\cdot))=0$ in $\\mathbb{R}^d\\setminus \\{ x_0 \\}$.\n\\end{proof}\n\nNext, we treat the case where $ 1\\le \\delta< \\infty$ and $|x-y|_\\infty < 4$.\n\n \\begin{thm}\\label{thm-f-1a}\n Suppose $ 1\\le \\delta<\\infty$. Then estimates \\eqref{f-2-0}, \\eqref{f-2-1} and \\eqref{f-2-00}\n continue to hold\n for $x, y\\in \\mathbb{R}^d$ with $|x-y|_\\infty< 4$.\n \\end{thm}\n\n\\begin{proof}\nThe proof is similar to that of Theorem \\ref{thm-f-1}.\nFix $x_0, y_0\\in\\mathbb{R}^d$ with $r=|x_0-y_0|_\\infty \\le 4$.\nLet $u$ be given by \\eqref{f-1} with $f\\in C_0^\\infty(Q(y_0, r); \\mathbb{R}^d)$.\nSince $\\mathcal{L}_\\delta (u)=0$ in $Q(x_0, r)$,\nin view of \\eqref{5.5-1} and Remark \\ref{re-large}, we obtain \n\\begin{equation}\\label{f-2-2a1}\n\\aligned\n|u(x_0)| + r |\\nabla u (x_0)|\n&\\le C \\left( \\fint_{Q(x_0, r)}\n|u|^2\\right)^{1\/2}\\\\\n& \\le C r^{1-\\frac{d}{2}} \\| f\\|_{L^p(\\mathbb{R}^d)}.\n\\endaligned\n\\end{equation}\nBy duality it follows that\n\\begin{equation}\\label{f-2-2a2}\n\\left(\\int_{Q(y_0, r)}\n|\\Gamma_\\delta (x_0, y)|^{p^\\prime}\\, dy \\right)^{1\/p^\\prime}\n\\le C r^{1-\\frac{d}{2}}.\n\\end{equation}\nSince $\\mathcal{L}_\\delta^* (\\Gamma_\\delta (x_0, \\cdot))\n=0$ in $Q(y_0, r)$, the desired estimates\nfollow readily from the first inequality in (\\ref{f-2-2a1}). \nWe omit the details.\n\\end{proof}\n\nIt remains to handle the case where $0<\\delta<1$ and $|x-y|_\\infty<4$.\n\n \\begin{lemma}\\label{lemma-f-2}\n Let $0< \\delta\\le 1$ and \n \\begin{equation}\\label{f-20-0}\n u(x)=\\int_{\\mathbb{R}^d} \\Gamma_\\delta (x, y) \\Lambda_\\delta (y) f(y)\\, dy\n \\end{equation}\n for some $f\\in C_0^\\infty(\\mathbb{R}^d; \\mathbb{R}^d)$.\n Then\n \\begin{equation}\\label{f-20-1}\n \\|\\Lambda_\\delta u\\|_{L^{p^\\prime} (\\mathbb{R}^d)}\n + \\| \\Lambda_\\delta \\nabla u \\|_{L^2(\\mathbb{R}^d)}\n \\le C \\| f\\|_{L^{p} (\\mathbb{R}^d)},\n \\end{equation}\n where $p =\\frac{2d}{d+2}$ and $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n \n\\begin{proof}\nAs in the proof of Lemma \\ref{lemma-f-1}, we have\n$$\n\\| u\\|_{L^{p^\\prime} (\\mathbb{R}^d)} \\le C \\| \\nabla u\\|_{L^2(\\mathbb{R}^d)} \n\\le C \\| D u \\|_{L^2(\\mathbb{R}^d)}\n$$\n and \n$\\| u\\|_{L^{p^\\prime} (\\omega)} +\\| \\nabla u\\|_{L^2(\\omega)} \\le C \\| Du \\|_{L^2(\\omega)}.\n$\nIt follows that \n\\begin{equation}\\label{f-2-3x}\n\\aligned\n\\| \\Lambda_\\delta u\\|_{L^{p^\\prime} (\\mathbb{R}^d)} \n+ \\|\\Lambda_\\delta \\nabla u\\|_{L^2(\\mathbb{R}^d)} \n & \\le C\\delta \\| Du \\|_{L^2(\\mathbb{R}^d)} \n+ C \\| Du \\|_{L^2(\\omega)}\\\\\n& \\le C \\|\\Lambda_\\delta D u \\|_{L^2(\\mathbb{R}^d)},\n\\endaligned\n\\end{equation}\nwhere we have used the assumption $\\delta\\le 1$ for the last inequality.\nBy letting $v=u$ in \\eqref{f-2}, we obtain \n$$\n\\|\\Lambda_\\delta Du \\|^2_{L^2(\\mathbb{R}^d)} \\le C \\| \\Lambda_\\delta u \\|_{L^{p^\\prime} (\\mathbb{R}^d)} \\| f\\|_{L^p(\\mathbb{R}^d)},\n$$\nwhich, together with \\eqref{f-2-3x}, yields \\eqref{f-20-1}.\n\\end{proof}\n\n\\begin{thm}\\label{thm-f-3}\nSuppose $0<\\delta< 1$.\nFor $x, y\\in \\mathbb{R}^d$ with $|x-y|_\\infty< 4 $, we have\n \\begin{align}\n \\Lambda_\\delta (x) \\Lambda_\\delta (y) |\\Gamma_\\delta (x, y)| & \\le\n C |x-y|^{2-d} , \\label{f-3-0}\\\\\n \\Lambda_\\delta (x) \\Lambda_\\delta (y) \n \\big\\{ |\\nabla_x \\Gamma_\\delta (x, y)|\n +|\\nabla _y \\Gamma_\\delta (x, y)| \\big\\} \n & \\le C |x-y|^{1-d} , \\label{f-3-1}\\\\\n \\Lambda_\\delta (x) \\Lambda_\\delta (y) |\\nabla_x \\nabla_y \\Gamma_\\delta (x, y)| \n & \\le C |x-y|^{-d}, \\label{f-3-1x} \n \\end{align}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$.\n \\end{thm}\n \n \\begin{proof}\n \n Fix $x_0, y_0\\in \\mathbb{R}^d$ with $r=|x_0-y_0|_\\infty<4$.\n Let $u$ be given by (\\ref{f-20-0}) with $f\\in C_0^\\infty(Q(y_0, r); \\mathbb{R}^d) $.\n Then $\\mathcal{L}_\\delta (u)=0$ in $Q(x_0, r)$.\n It follows from (\\ref{5.5-0}) that\n \\begin{equation}\\label{f-3-2}\n \\aligned\n \\Lambda_\\delta (x_0)\n \\big\\{ |u(x_0)| + r|\\nabla u(x_0) | \\big\\}\n &\\le C \\left(\\fint_{Q(x_0, r)}\n |\\Lambda_\\delta u|^2 \\right)^{1\/2}\\\\\n &\\le Cr^{1-\\frac{d}{2}}\n \\| f\\|_{L^p(\\mathbb{R}^d)},\n \\endaligned\n \\end{equation}\n where we have used \\eqref{f-20-1} for the last inequality.\n By duality this implies that\n \\begin{equation}\\label{f-3-3}\n \\Lambda_\\delta (x_0)\n \\left(\\int_{Q(y_0, r)}\n |\\Lambda_\\delta (y) \\Gamma_\\delta (x_0, y)|^{p^\\prime} \\, dy \\right)^{1\/p^\\prime}\n \\le C r^{1-\\frac{d}{2}}.\n \\end{equation}\n Since $\\mathcal{L}_\\delta^* ( \\Gamma_\\delta (x_0, \\cdot)) =0$ in $\\mathbb{R}^d\\setminus \\{ x_0\\}$, we may use the first inequality in \\eqref{f-3-2}\n and (\\ref{f-3-3}) \n to obtain\n \\begin{equation}\\label{f-3-4}\n \\Lambda_\\delta (x_0)\\Lambda_\\delta (y_0)\n |\\Gamma_\\delta (x_0, y_0)|,\n \\le C r^{2-d}.\n \\end{equation}\n which gives \\eqref{f-3-0}.\n The estimates in \\eqref{f-3-1} also follow from the first inequality in \\eqref{f-3-2}\n and (\\ref{f-3-3}). \n Finally, \\eqref{f-3-1x} follows from the first inequality in \\eqref{f-3-2} and \\eqref{f-3-1}.\n \\end{proof}\n \nWe end this section with a decay estimate of $D\\Gamma_\\delta (x, y)$ for $x\\in F$, as $\\delta \\to \\infty$.\n \n \\begin{thm}\\label{thm7.2}\n Let $1\\le \\delta<\\infty$.\n Let $u\\in H^1(Q(x_0, R); \\mathbb{R}^d)$ be a weak solution of\n $\\mathcal{L}_\\delta (u) =0$ in $Q(x_0, R)$\n for some $x_0\\in F$ and $R\\ge 5$.\n Then\n \\begin{equation}\\label{7.2-0}\n |Du(x_0)|\n \\le \\frac{C}{\\delta^2}\n \\left(\\fint_{Q(x_0, R)} | D u|^2\\, dx \\right)^{1\/2},\n \\end{equation}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $\\omega$, and $(\\sigma, M_0)$.\n \\end{thm}\n \n \\begin{proof}\nSuppose $x_0\\in F_k\\subset Q(x_0, 2)$ for some $k$.\nIt follows from \\eqref{local-2-0b} and interior estimates that\n\\begin{equation}\\label{7.2-1}\n|\\nabla u(x_0)|\n\\le C\\delta^{-2} \\left(\\int_{Q(x_0, 2)} |\\nabla u|^2 \\, dx \\right)^{1\/2}\n+ C \\left(\\int_{F_k} |\\nabla u|^2 \\, dx \\right)^{1\/2}.\n\\end{equation}\nChoose $\\phi\\in \\mathcal{R}$ such that $u-\\phi \\perp \\mathcal{R}$ in $H^1(F_k; \\mathbb{R}^d)$.\nSince $u-\\phi$ satisfies the same conditions as $u$, we may use \\eqref{7.2-1} with $u-\\phi$ in the place of $u$.\nAs a result, \n$$\n\\aligned\n|Du(x_0)| & \\le |\\nabla (u-\\phi) (x_0) |\\\\\n& \\le C\\delta^{-2} \\left(\\int_{Q(x_0, 2)} |\\nabla u|^2 \\, dx \\right)^{1\/2}\n+ C \\left(\\int_{F_k} | D u|^2 \\, dx \\right)^{1\/2},\n\\endaligned\n$$\nwhere we have used the second Korn inequality as well as the fact \n$|\\nabla \\phi|\\le C \\|\\nabla u\\|_{L^2(F_k)}$.\nThis, together with \\ref{7.1-0} with $f=0$, gives\n\\begin{equation}\\label{7.2-2}\n\\aligned\n|D u(x_0)|\n & \\le C \\delta^{-2}\n\\left (\\fint_{Q(x_0, 5)} |\\nabla u|^2\\right)^{1\/2}\\\\\n& \\le C \\delta^{-2}\n\\left (\\fint_{Q(x_0, R)} |\\nabla u|^2\\right)^{1\/2},\n\\endaligned\n\\end{equation}\nwhere we have used Theorem \\ref{main-thm-1} for the last inequality.\nChoose $\\psi$ in $\\mathcal{R}$ so that\n$$\n\\| \\nabla (u-\\psi)\\|_{L^2(Q(x_0, R)) } \\le C \\| D u \\|_{L^2(Q(x_0, R))}.\n$$\nIt follows that\n$$\n\\aligned\n|Du (x_0)|\n & = |D (u-\\psi) (x_0)|\\\\\n & \\le C \n \\delta^{-2}\n\\left (\\fint_{Q(x_0, R)} |\\nabla (u-\\psi) |^2\\right)^{1\/2}\n\\le C\\delta^{-2}\n \\left (\\fint_{Q(x_0, R)} | D u|^2\\right)^{1\/2}.\n\\endaligned\n$$\nwhere we have used \\eqref{7.2-2} with $u-\\psi$ in the place of $u$.\n \\end{proof}\n \n \\begin{cor}\n Let $1\\le \\delta<\\infty$. Then \n \\begin{equation}\\label{7.3-0}\n\\Lambda_{\\delta^2} (x) \n|D_x \\Gamma_\\delta (x, y)|\n+ \\Lambda_{\\delta^2} (y)\n|D_y \\Gamma_\\delta (x, y)| \n\\le C |x-y|^{1-d}\n\\end{equation}\nfor any $x, y\\in \\mathbb{R}^d$ with $|x-y|_\\infty\\ge 4$.\n \\end{cor}\n \n \\begin{proof}\nSince $\\delta\\ge 1$, it follows by Theorems \\ref{thm-f-1} and \\ref{thm-f-1a} that\n$$\n|\\nabla_x \\Gamma_\\delta (x, y)| + |\\nabla_y \\Gamma_\\delta (x, y)| \\le\nC |x-y|^{1-d}.\n$$\nfor any $x, y\\in \\mathbb{R}^d$ and $x\\neq y$.\n This, together with Theorem \\ref{thm7.2}, gives \\eqref{7.3-0}.\n \\end{proof}\n \n\n \n \\section{Lipschitz estimates for $\\mathcal{L}_\\delta (u)=f$} \\label{section-7}\n \n The goal of this section is to prove \\eqref{f-Lip-f}.\n The case $0<\\delta<\\infty$ follows readily from Theorem \\ref{main-thm-2} and estimates of fundamental solutions in Section \\ref{section-6}.\n To handle the cases $\\delta=0$ and $\\delta =\\infty$, we use an approximation argument.\n \n Let $\\Omega$ be a bounded Lipschitz domain in $\\mathbb{R}^d$.\n We call $\\Omega$ a type II domain (with respect to $\\omega$) if $F_k \\cap \\Omega\\neq \\emptyset$ implies that \n $\\widetilde{F}_k \\subset \\Omega$.\n In particular, if $\\Omega$ is a type II Lipschitz domain, then $\\partial \\Omega \\cap \\partial \\omega =\\emptyset$.\n \n \\begin{lemma}\\label{lemma-app-1}\nAssume that $A$ and $\\omega$ satisfy the same conditions as in Theorem \\ref{main-thm-1}.\n Let $0<\\delta\\le 1$ and $\\Omega$ be a type II Lipschitz domain.\n Let $u_\\delta \\in H^1(\\Omega; \\mathbb{R}^d)$ be a weak solution of\n $\\mathcal{L}_\\delta (u_\\delta)=f \\chi_\\omega $ in $\\Omega$ and $u_0\\in H^1(\\Omega; \\mathbb{R}^d)$ a weak solution of\n $\\mathcal{L}_0 (u_0)=f\\chi_\\omega $ in $\\Omega$, where $f\\in L^2(\\Omega; \\mathbb{R}^d)$.\n Suppose that $u_\\delta=u_0$ on $\\partial\\Omega$.\n Then\n \\begin{equation}\\label{app-1-0}\n \\| u_\\delta -u_0\\|_{H^1(\\Omega)}\n \\le C \\delta \\| D u_0\\|_{L^2(\\Omega\\cap F)},\n \\end{equation}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n \n \\begin{proof}\n Let $w=u_\\delta -u_0\\in H^1_0(\\Omega; \\mathbb{R}^d) $.\n Since $u_0$ is a weak solution of $\\mathcal{L}_0 (u_0)=f\\chi_\\omega $ in $\\Omega$,\n \\begin{equation}\\label{app-1-1}\n \\int_{\\Omega\\cap \\omega} A\\nabla u_0 \\cdot \\nabla v \\, dx\n =\\int_{\\Omega\\cap \\omega} f \\cdot v\\, dx\n \\end{equation}\n for any $v\\in H^1_0(\\Omega; \\mathbb{R}^d)$. We also assume that $-\\text{\\rm div}(A\\nabla u_0)=0$ in $F\\cap \\Omega$.\n Using\n $$\n \\int_{\\Omega\\cap \\omega} A\\nabla u_\\delta \\cdot \\nabla v\\, dx\n + \\delta^2 \\int_{\\Omega\\cap F} A\\nabla u_\\delta \\cdot \\nabla v\\, dx\n =\\int_{\\Omega\\cap \\omega} f\\cdot v\\, dx,\n $$\n we obtain \n $$\n \\int_{\\Omega\\cap \\omega}\n A\\nabla w \\cdot \\nabla w\\, dx\n + \\delta^2 \\int_{\\Omega\\cap F} A \\nabla w \\cdot \\nabla w \n =-\\delta^2 \\int_{\\Omega\\cap F} A\\nabla u_0 \\cdot \\nabla w\\, dx.\n $$\n Hence, by \\eqref{ellipticity} and the Cauchy inequality, \n \\begin{equation}\\label{app-1-2}\n \\int_{\\Omega\\cap \\omega}\n |D w|^2\\, dx\n + \\delta^2 \\int_{\\Omega\\cap F} |D w|^2\\, dx\n \\le C \\delta^2 \\int_{\\Omega\\cap F} |D u_0|^2\\, dx.\n \\end{equation}\n Note that $\\text{\\rm div}(A\\nabla w)=0$ in $F_k$ for any $F_k\\subset \\Omega$.\n By Lemma \\ref{lemma-2.2} we have \n $$\n \\| D w\\|_{L^2(F_k)}\\le C \\| D w \\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}.\n $$\n As a result, $\\| Dw\\|_{L^2(\\Omega\\cap F)}\\le C \\| Dw \\|_{L^2(\\Omega\\cap \\omega)}$,\n where we have used the assumption that $\\Omega$ is a type II Lipschitz domain.\n This, together with \\eqref{app-1-2} and the first Korn inequality \\cite[p.13]{OSY-1992}, gives \\eqref{app-1-0}.\n \\end{proof}\n \n \\begin{lemma}\\label{lemma-app-2}\n Assume that $A$ and $\\omega$ satisfy the same conditions as in Theorem \\ref{main-thm-1}.\n Let $1\\le \\delta<\\infty$ and $\\Omega$ be a type II Lipschitz domain.\n Let $u_\\delta \\in H^1(\\Omega; \\mathbb{R}^d)$ be a weak solution of\n $\\mathcal{L}_\\delta (u_\\delta)=f $ in $\\Omega$ and $u_\\infty\\in H^1(\\Omega; \\mathbb{R}^d)$ a weak solution of\n $\\mathcal{L}_\\infty (u_\\infty)=f$ in $\\Omega$, where $f\\in L^2(\\Omega; \\mathbb{R}^d)$.\n Suppose that $u_\\delta=u_\\infty$ on $\\partial\\Omega$.\n Then\n \\begin{equation}\\label{app-2-0}\n \\| u_\\delta -u_\\infty \\|_{H^1(\\Omega)}\n \\le C \\delta^{-1} \\big\\{ \n \\| D u_\\infty\\|_{L^2(\\Omega\\cap \\omega)}\n + \\| f\\|_{L^2(\\Omega)} \\big\\},\n \\end{equation}\n where $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, and $\\omega$.\n \\end{lemma}\n\n\\begin{proof}\nSince $u_\\infty$ is a weak solution of $\\mathcal{L}_\\infty (u_\\infty) =f$, it follows that \n$$\n\\int_{\\Omega\\cap \\omega} A\\nabla u_\\infty \\cdot \\nabla v\\, dx\n=\\int_\\Omega f \\cdot v\\, dx\n$$\nfor any $v\\in H^1_0(\\Omega; \\mathbb{R}^d)$ with $Dv=0$ in $\\Omega\\cap F$, and that \n$D u_\\infty=0$ in $\\Omega\\cap F$.\nLet $\\phi \\in H^1_0(\\Omega; \\mathbb{R}^d)$.\nFor each $F_k \\subset \\Omega$ and $g_k \\in \\mathcal{R}$, let $w_k \\in H^1_0(\\widetilde{F}_k; \\mathbb{R}^d)$ be an extension \nof $\\phi-g_k$ from $F_k$ to $\\widetilde{F}_k$ with the property that\n$$\n\\| w_k \\|_{H^1(\\widetilde{F}_k)} \\le C \\| \\phi- g_k\\|_{H^1(F_k)}.\n$$\nExtend $w_k$ from $\\widetilde{F}_k$ to $\\mathbb{R}^d$ by zero, and define \n$$\nv=\\phi -\\sum_k w_k, \n$$\nwhere the sum is taken over those $k$'s for which $F_k\\subset \\Omega$.\nNote that $v\\in H^1_0(\\Omega; \\mathbb{R}^d)$ and $v =g_k$ on $F_k$. Since $\\Omega$ is a type II domain,\nit follows that $Dv =0$ on $\\Omega\\cap F$.\nAs a result, we see that \n$$\n\\aligned\n & \\Big|\n\\int_{\\Omega\\cap \\omega} A \\nabla u_\\infty\\cdot \\nabla \\phi\\, dx -\\int_{\\Omega} f \\cdot \\phi \\, dx \\Big|\\\\\n&\n= \\Big| \\int_{\\Omega\\cap \\omega} A \\nabla u_\\infty\\cdot \\sum_k \\nabla w_k \\, dx -\\int_{\\Omega} f \\cdot \\sum_k w_k \\, dx \\Big|\\\\\n&\\le C \\sum_k \\| D u_\\infty\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n\\| D w_k \\|_{L^2(\\widetilde{F}_k)}\n+ C \\sum_k \\| f\\|_{L^2(\\widetilde{F}_k)}\n\\| w_k \\|_{L^2(\\widetilde{F}_k)}\\\\\n&\\le C \\sum_k\n\\big( \\| D u_\\infty\\|_{L^2(\\widetilde{F}_k \\setminus \\overline{F}_k)}\n+ \\| f\\|_{L^2(\\widetilde{F}_k)}\\big) \\| \\phi -g_k \\|_{H^1(F_k)}\\\\\n& \\le C \\big( \\| D u_\\infty\\|_{L^2(\\Omega\\cap \\omega)}\n+\\| f\\|_{L^2(\\Omega)} \\big) \\| D \\phi\\|_{L^2(\\Omega\\cap F)},\n\\endaligned\n$$\nwhere we have chosen $g_k \\in \\mathcal{R}$ so that\n$\\| \\phi - g_k \\|_{H^1(F_k)} \\le C \\| D \\phi\\|_{L^2(F_k)}$.\nThis, together with \n$$\n\\int_{\\Omega\\cap \\omega} A\\nabla u_\\delta \\cdot \\nabla \\phi\\, dx\n+ \\delta^2 \\int_{\\Omega\\cap F} A \\nabla u_\\delta \\cdot \\nabla \\phi\\, dx\n=\\int_\\Omega f \\cdot \\phi, dx,\n$$\nimplies that\n$$\n\\aligned\n & \\Big| \\int_{\\Omega\\cap \\omega} A\\nabla (u_\\delta -u_\\infty) \\cdot \\nabla \\phi\\, dx\n+ \\delta^2 \\int_{\\Omega\\cap F} A \\nabla (u_\\delta -u_\\infty) \\cdot \\nabla \\phi\\, dx \\Big|\\\\\n&\\le \nC \\big( \\| D u_\\infty\\|_{L^2(\\Omega\\cap \\omega)}\n+\\| f\\|_{L^2(\\Omega)} \\big) \\| D \\phi\\|_{L^2(\\Omega\\cap F)}.\n\\endaligned\n$$\nBy letting $w=u_\\delta -u_\\infty$ and $ \\phi=w$ in the inequality above, we obtain\n$$\n\\| D w \\|_{L^2(\\Omega\\cap \\omega)}\n+\\delta \\| D w\\|_{L^2(\\Omega\\cap F)}\n\\le C \\delta^{-1} \\big\\{ \\| D u_\\infty\\|_{L^2(\\Omega\\cap \\omega)}\n+ \\| f\\|_{L^2(\\Omega)} \\big\\},\n$$\nwhere we have also used the Cauchy inequality.\nSince $\\delta\\ge 1$ and $w\\in H^1_0(\\Omega; \\mathbb{R}^d)$, the estimate \\eqref{app-2-0}\nfollows by the first Korn inequality.\n\\end{proof}\n\n\\begin{thm}\\label{main-thm-3}\nLet $ d\\ge 3$.\nAssume that $A$ satisfies \\eqref{ellipticity}, \\eqref{smoothness}, and is 1-periodic.\nAlso assume that each $F_k$ is a bounded $C^{1, \\sigma}$ domain.\n\n\\begin{enumerate}\n\n\\item\n\nLet $0\\le \\delta < 1$.\nLet $u\\in H^1(\\Omega; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)=f\\chi_\\omega$ in $Q(x_0, R)$ for some $R\\ge 6$, where $f\\in L^p(Q(x_0, R); \\mathbb{R}^d)$ for some $p>d$.\nThen\n\\begin{equation}\\label{lip-f-8}\n|\\nabla u (x_0)|\n\\le C \\left(\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2}\n+C R \\left(\\fint_{Q(x_0, R)} |f|^p \\right)^{1\/p},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $p$, $\\omega$, and $(\\sigma, M_0)$ in \\eqref{smoothness}.\n\n\\item\n\nLet $1\\le \\delta \\le \\infty$.\nLet $u\\in H^1(\\Omega; \\mathbb{R}^d)$ be a weak solution of\n$\\mathcal{L}_\\delta (u)= f$ in $Q(x_0, R)$ for some $R\\ge 6$, where $f\\in L^p(Q(x_0, R); \\mathbb{R}^d)$ for some $p>d$.\nThen\n\\begin{equation}\\label{lip-f-9}\n|\\nabla u (x_0)|\n\\le \\left(\\fint_{Q(x_0, R)\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2}\n+C R \\left(\\fint_{Q(x_0, R)} |f|^p \\right)^{1\/p},\n\\end{equation}\nwhere $C$ depends only on $d$, $\\kappa_1$, $\\kappa_2$, $p$, $\\omega$, and $(\\sigma, M_0)$.\n\n\\end{enumerate}\n\\end{thm}\n\n \\begin{proof}\n \n By translation we may assume $x_0=0$.\n We consider 4 cases.\n \n \\medskip\n \n \\noindent{\\bf Case 1.} Assume $1\\le \\delta < \\infty$.\n If $f=0$, this is given by Theorem \\ref{main-thm-2}.\n In general, let\n \\begin{equation}\\label{lip-f-10}\n v(x) =\\int_{Q_R} \\Gamma_\\delta (x, y) f(y)\\, dy.\n \\end{equation}\n Then $\\mathcal{L}_\\delta (v)=f$ in $Q_R$, and by Theorems \\ref{thm-f-1} and \\ref{thm-f-1a},\n \\begin{equation}\\label{lip-f-11}\n \\|\\nabla v\\|_{L^\\infty (Q_R)} \\le CR \\left(\\fint_{Q_R} |f|^p\\right)^{1\/p}\n \\end{equation}\nfor $p>d$. Hence,\n $$\n \\aligned\n |\\nabla u(0)|\n &\\le |\\nabla (u-v)(0)| +|\\nabla v(0)|\\\\\n & \\le C \\left(\\fint_{Q_R\\cap \\omega } |\\nabla (u-v)|^2\\right)^{1\/2}\n + C R \\left(\\fint_{Q_R} |f|^p\\right)^{1\/p}\\\\\n &\\le C \\left(\\fint_{Q_R\\cap \\omega } |\\nabla u|^2 \\right)^{1\/2}\n+C R \\left(\\fint_{Q_R } |f|^p \\right)^{1\/p},\n\\endaligned\n$$\nwhere we have used the fact $\\mathcal{L}_\\delta (u-v)=0$ in $Q_R$.\n \n \\medskip\n \n \\noindent{\\bf Case 2.} \n Assume $\\delta=\\infty$.\n In this case we use an approximation argument.\n Choose a type II Lipschitz domain $\\Omega$ such that $Q_{R-2}\\subset \\Omega\\subset Q_R$.\n Let $u_\\delta\\in H^1(\\Omega: \\mathbb{R}^d)$ be a weak solution of $\\mathcal{L}_\\delta (u_\\delta)=f$ in $\\Omega$\n such that $u_\\delta=u$ on $\\partial\\Omega$.\n It follows by Lemma \\ref{lemma-app-2} that $u_\\delta \\to u$ in $H^1(\\Omega; \\mathbb{R}^d)$, as $\\delta \\to \\infty$.\n By the proof for Case 1, \n $$\n \\left(\\fint_{Q_r} |\\nabla u_\\delta|^2\\right)^{1\/2}\n \\le C \\left(\\fint_{Q_{R-2}\\cap \\omega } |\\nabla u_\\delta |^2 \\right)^{1\/2}\n+C R \\left(\\fint_{Q_R} |f|^p \\right)^{1\/p},\n $$\n for $r\\in (0, 1\/4)$.\n The proof is complete by letting $\\delta\\to \\infty$ and then $r\\to 0$ in the inequality above.\n \n \\medskip\n \n \\noindent{\\bf Case 3.}\n Assume $0<\\delta< 1$.\n If $f=0$, the estimate \\eqref{lip-f-8} is given by Theorem \\ref{main-thm-2}.\n In general, let\n $$\n v(x)=\\int_{Q_R\\cap \\omega} \\Gamma_\\delta (x, y) f(y)\\, dy.\n $$\n Then\n $\n \\mathcal{L}_\\delta (v)=f\\chi_\\omega$ in $Q_R$, and by Theorems \\ref{thm-f-1} and \\ref{thm-f-3},\n $$\n \\|\\Lambda_\\delta \\nabla v\\|_{L^\\infty(Q_R)}\n \\le C R \\left(\\fint_{Q_R} |f|^p \\right)^{1\/p}.\n $$\n Observe that since $\\text{\\rm div}(A\\nabla v)=0$ in $Q_R\\cap F$, it follows from Theorem \\ref{local-thm-0} that\n $$\n \\aligned\n |\\nabla v (0)|\n & \\le C \\left( \\fint_{Q_1} |\\nabla v|^2 \\right)^{1\/2}\n + C \\left(\\fint_{Q_1} |f|^p \\right)^{1\/p}\\\\\n& \\le C \\left(\\fint_{Q_4\\cap \\omega} |\\nabla v|^2 \\right)^{1\/2}\n + C \\left(\\fint_{Q_1} |f|^p \\right)^{1\/p},\n\\endaligned\n$$\n where we have used \\eqref{2.2-0} for the last inequality.\n Hence,\n $$\n \\aligned\n |\\nabla u (0)|\n &\\le |\\nabla (u-v)(0)| + |\\nabla v(0)|\\\\\n &\\le C \\left(\\fint_{Q_R\\cap \\omega} |\\nabla (u-v)|^2\\right)^{1\/2}\n + C R \\left(\\fint_{Q_R} |f|^p \\right)^{1\/p}\\\\\n & \\le C \\left(\\fint_{Q_R\\cap \\omega} |\\nabla u|^2\\right)^{1\/2}\n +C R \\left(\\fint_{Q_R} |f|^p \\right)^{1\/p}.\n \\endaligned\n $$\n \n \\noindent{\\bf Case 4.}\n Assume $\\delta=0$.\n As in Case 2, this follows from Case 3 by using the approximation in Lemma \\ref{lemma-app-2}.\n We omit the details.\n\\end{proof} \n \n \\bibliographystyle{amsplain}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n \\label{intro}\n\nBlack hole X-ray binaries (BHBs) are known to occupy distinct spectral states\nwhich can be characterized by the relative contribution of thermal and\nnon-thermal emission components (e.g. McClintock \\& Remillard 2004). The most\nwell-understood of these states is the thermal dominant (or high\/soft) state.\nHere most of the flux is in the thermal component, which is generally assumed to\nbe emission from a radiatively efficient, geometrically thin accretion disk\n(Shakura \\& Sunyaev 1973). The disk is believed to extend deep within the\ngravitational field of the black hole, and this makes spectral modeling of\nthis state an important probe of both the physics of relativistic accretion\ndisks and the properties of black holes.\n\nIn standard treatments of black hole accretion disks, the emitting matter\nextends down to the innermost stable circular orbit (ISCO), which is determined\nby the mass and spin of the black hole. Such treatments usually assume a\n``no-torque'' inner boundary condition at this radius (e.g. Novikov\n\\& Thorne 1973), but magnetic fields may in fact exert such torques\n(Gammie 1999; Krolik 1999; Hawley \\& Krolik 2002), increasing the radiative\nefficiency of the disk. The structure and emission of these disks are therefore\nsensitive to the mass and spin of the black hole as well as any torque which may\nbe present.\n\nThis sensitivity makes accretion disk spectral modeling a potential way to\nmeasure or constrain black hole spin (e.g. Ebisawa et al. 1993; Zhang et al. 1997;\nShafee et al. 2006; Middleton et al. 2006).\nThis method requires a model of the radial profile of\neffective temperature in the gravitational field of the black hole, calculation\nof the relativistic transfer function from the disk surface to an observer at\ninfinity (Cunningham 1975), and spectral modeling of the surface emission in the\nlocal rest frame of the disk. Most implementations of this method approximate\none or all of these components. One common approximation is to assume that the\ndisk surface emission is a blackbody or, more generally, a color-corrected\nblackbody,\n \\begin{eqnarray} \nI_{\\nu}=f^{-4}B_{\\nu}(fT_{\\rm eff}), \\label{e:ccbb} \n \\end{eqnarray} \nwhere $I_{\\nu}$ is the specific intensity, $T_{\\rm eff}$ is the effective\ntemperature, $B_{\\nu}$ is the Planck function, and $f$ is the spectral hardening\nfactor (or color-correction), typically assumed to be around 1.7 \n(Shimura \\& Takahara 1995). One of\nthe most sophisticated models of this type is the KERRBB model (Li et al. 2005)\nfor Xspec (Arnaud 1996). It accounts for the relativistic effects on the disk\neffective temperature profile and the relativistic transfer function. \nPotential difficulties with the color-corrected blackbody approximation exist.\nThe local spectrum may not be well approximated by an isotropic, color-corrected\nblackbody due to limb darkening and frequency dependent absorption opacities. \nEven if it can, one must still specify $f$.\nIt has been suggested that $f$ is a relatively strong function\nof accretion rate or of the fraction of energy emitted in a corona (Merloni et\nal. 2000).\n\nRelativistic models (Davis et al. 2005) now exist which calculate\nthe non-LTE vertical disk structure and radiative transfer self-consistently\nusing the TLUSTY stellar atmospheres code (Hubeny \\& Lanz 1995). The\nrelativistic effects on photon geodesics are accounted for with ray tracing\nmethods (Agol 1997). These spectral models have now been implemented in an\nXspec table model (BHSPEC; Davis \\& Hubeny, 2006). By calculating\nvalues of $f$ appropriate for use with KERRBB, these models have already been\nused to estimate black hole spins from KERRBB fits to two BHBs: GRO J1655-40\nand 4U 1543-47 (Shafee et al. 2006).\n\nIn this work, we circumvent the color-corrected blackbody approximation\nentirely by fitting the BHSPEC model directly to BHB observations. Since the\nmodel does not include irradiation of the disk surface, we focus our efforts\non thermal dominant state observations in which the non-thermal emission is a small\nfraction of the total flux. Fortunately, a sample of such observations made\nwith the {\\it Rossi X-ray Timing Explorer} ({\\it RXTE}) already exists\n(Gierlinski \\& Done 2004; hereafter GD04).\n\nBHSPEC still assumes that the disk emission extends only down to the ISCO\nand then effectively ceases due to rapid in-fall of matter interior to this\nradius. This assumption appears to be consistent with spectral modeling of\nBHBs in the thermal dominant state in that the luminosity $L$ is seen to scale roughly\nwith the fourth power of the color temperature $T_{\\rm c}$ in several different\nsources (Kubota et al. 2001; Gierlinski \\& Done 2004). This\nsuggests that as sources vary by over an order of magnitude in luminosity,\nthere is a roughly constant emitting area and thus a relatively constant inner\nradius to the disk. It is therefore very natural to associate such a stable\ninner radius with the ISCO of the black hole, though the `emission edge'\nneed not coincide exactly with the ISCO (Krolik \\& Hawley 2002).\n\nNot all of these sources follow the $L \\propto T_{\\rm c}^4$ relation exactly,\nhowever. In several cases a relative hardening is seen \nwith increasing $L$ (see e.g. GD04; Kubota \\& Makishima 2004; Shafee et al. 2006).\nA potential explanation for this hardening is that \nadvection may be becoming increasingly important as these sources approach \nthe Eddington limit (Kubota \\& Makishima 2004).\nAlternatively, this behavior is qualitatively consistent with the increased \nspectral hardening with accretion rate in the local disk atmospheres, ignoring \nadvection (Davis et al. 2005; Shafee et al. 2006). In\nthe BHSPEC model, the precise nature of the hardening depends strongly on the\nvariation in surface density with radius in the disk. Currently, the surface\ndensity is determined by assuming the vertically averaged stress is proportional\nto the vertically averaged total pressure with a constant of proportionality\n$\\alpha$. However, more general stress prescriptions could be implemented in the\nfuture. Therefore, spectral modeling could potentially provide a constraint on\nthe nature of the stresses in these systems.\n\nIn this paper we fit these fully-relativistic, non-LTE accretion disk models to\n{\\it RXTE} and {\\it BeppoSAX} observations of BHBs in the thermal dominant state.\nOur purpose is two-fold: we want to test the applicability of the spectral\nmodels to these observations, and having then found suitable representations\nof the data, we use them to infer black hole spins and infer properties of the\nstresses in these system.\nWe review our spectral fitting and results in section \\ref{specmod}, discuss the\nimplications of these results in section \\ref{discus}, and summarize our\nconclusions in section \\ref{conc}. In the Appendix we develop a simplified model\nto understand and motivate the variation of the BHSPEC spectra with accretion rate.\n\n\n\\begin{figure}\n\\includegraphics[width=0.46\\textwidth]{f1.eps}\n\\caption{The unfolded spectrum for the {\\it BeppoSAX}\nobservation of LMC X-3 using the best fit BHSPEC models for $i=67^{\\circ}$, \n$D=52$ kpc, and $\\alpha=0.01$. The total model component (green, solid curve),\nBHSPEC (red, long-dashed curve) and COMPTT (violet, short-dashed curve) are\nplotted. The unabsorbed BHSPEC model (orange, solid curve) is also shown.\n\\label{fig1}}\n\\end{figure}\n\n\\section{Spectral Modeling}\n\\label{specmod}\n\nSeveral properties of BHBs make them particularly well suited both for testing\naccretion disk theory and for measuring the unknown black hole properties of\ninterest. There are several sources for which precise, independent\nmeasurements of the mass of the primary and \nthe binary inclination are\navailable from light curve modeling of the secondary star (e.g. Orosz \\& Bailyn\n1997). Typically, the distance to BHBs are known with less precision, but there\nare exceptions. As can be seen in Table \\ref{tbl1}, the distances to LMC X-3 and\nGRO J1655-40 are both claimed to be known to better than 10\\%. \nBHBs have an advantage over most active galactic nuclei because of their relatively\nshort timescales\nfor large changes in the bolometric luminosity, which allows the same source to\nbe observed in the same spectral state at appreciably different accretion\nrates. The variation of the BHSPEC model with accretion rate provides a precise\nquantitative prediction which is sensitive to the assumed stress prescription. \nTherefore, simultaneous fitting of multiple observations of the same source at\ndifferent epochs provides a much more powerful constraint and potentially more\ninformation than a single fit to a single epoch.\n\nWe compare three accretion disk models for the soft, thermal emission: the\nmulticolor disk model (DISKBB in Xspec, Mitsuda et al. 1984), KERRBB, and BHSPEC.\nDISKBB is the most commonly used spectral model, but it neglects relativity. \nBoth KERRBB and BHSPEC include relativistic effects. In addition, BHSPEC includes \natmosphere physics which we are interested in testing in this paper. In particular,\nwe are interested in examining whether BHSPEC with a fixed $\\alpha$ can explain the\nvariation in the spectral hardening with accretion rate. As a control, we compare\nthese results with KERRBB at a fixed $f$.\n\nWe also model the neutral absorption along the line of sight, and though we focus\non observations inferred to be disk dominated, we need an additional component to\naccount for the non-thermal emission which is present. It is widely believed\nthat the coronal emission is due to inverse Compton scattering of seed photons\nfrom the accretion disk. In this case, a power law will tend to overestimate the\nflux at low energies. Therefore, we prefer to approximate the non-thermal\nemission with the COMPTT Comptonization model (Titarchuk 1994) in our spectral\nfits. We specify a disk geometry and fix the high energy cutoff in this model to\n50 keV, a value high enough not to affect our fits. We also tie the seed photon\ntemperature to the DISKBB model temperature\nwhen DISKBB is used to model the soft emission. For KERRBB and BHSPEC we fix\nthis parameter at the best fit DISKBB value. This leaves two free parameters --\nan optical depth and a normalization for each data set.\n\nDISKBB is a relatively simple model with only two parameters: the temperature at\nthe inner edge of the disk $T_{\\rm in}$ and the model normalization. BHSPEC and\nKERRBB both share a number of model parameters which need to be specified or fit.\nThe black hole spin $a_{\\ast}\\equiv a\/M$ and the accretion rate $\\dot{M}$ are free\nparameters in all fits unless stated otherwise. For BHSPEC $\\dot{M}$ is\nparameterized by $\\ell \\equiv L\/L_{\\rm Edd}$ where $L_{\\rm Edd}$ is the Eddington\nluminosity for completely ionized hydrogen. This value is converted to an accretion\nrate by assuming an efficiency $\\eta$ which corresponds to the fraction of\ngravitational binding energy at infinity which is converted to radiation. Other\nparameters include black hole mass $M$, disk inclination $i$, and distance to the\nsource $D$. For KERRBB $D$ is an explicit parameter, but for BHSPEC\n$D=10\/\\sqrt{N}$ kpc where $N$ is the model normalization.\nIn some cases $M$, $i$, and $D$ (or $N$) are fixed at the estimates\ngiven in Table \\ref{tbl1}. Though $M$ is always fixed, there are also cases\nwhere $i$ and $D$ are left as free parameters. The estimates for $i$ in Table\n\\ref{tbl1} are all estimates of the binary inclination. However, there is\nno guarantee that the angular momentum of the black hole, and therefore the inner\naccretion disk, is aligned with the binary (Bardeen \\& Petterson 1975). If one\nassumes that the jet axis is aligned with the angular momentum vector of the\nblack hole, then misalignments may be common (Maccarone 2002). XTE J1550-564\nand GRO J1655-40 are among the sources for which misalignment can be inferred. \nTherefore, we also consider fits in which $i$ and $D$ are free parameters with\n$i$ unconstrained and $D$ allowed to vary within the confidence intervals in\nTable \\ref{tbl1}. These models also allow for the presence of a torque on the\ninner disk which is parameterized by the increase in efficiency due to the torque\nrelative to the efficiency of the untorqued disk $\\Delta \\eta\/\\eta$.\n\n\\begin{deluxetable*}{lccccl}\n\\tablecolumns{6}\n\\tablecaption{Source Descriptions\\label{tbl1}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Name} &\n\\colhead{Mass} &\n\\colhead{Distance} &\n\\colhead{Inclination} &\n\\colhead{$N_{\\rm H}$} &\n\\colhead{Reference}\\\\\n &\n\\colhead{$( M_{\\odot})$} &\n\\colhead{(kpc)} &\n\\colhead{(deg)} &\n\\colhead{$(10^{22}\\,{\\rm cm}^{-2})$} &\n}\n\n\\startdata\nLMC X-3 &\n$7 (5-11)$ &\n$52 (51.4-52.6)$ &\n$67 (65-69)$ &\n$0.04$ &\n1, 2, 3, 4, 5 \\\\\nXTE J1550-564 &\n$10 (9.7-11.6)$ &\n$5.3 (2.8-7.6)$ &\n$72 (70.8-75.4)$ &\n$0.65$ &\n6, 7 \\\\\nGRO J1655-40 &\n$7 (6.8-7.2)$ &\n$3.2 (3.0-3.4)$ &\n$70 (64-71)$ &\n$0.8$ &\n8, 9, 10, 11\\\\\n\\enddata\n\n\\tablerefs{(1) Soria et al. (2001); (2) Cowley et al. (1983); (3) di Benedetto (1997); \n(4) Kuiper et al. (1988); (5) Page et al. (2003); (6) Orosz et al. (2002);\n(7) Gierli\\'nski \\& Done (2003); (8) Shahbaz et al. (1999);\n(9) Hjellming \\& Rupen (1995); (10) van der Hooft et al. (1998);\n(11) Gierli\\'nski et al. (2001)}\n\n\\end{deluxetable*}\n\nThere are other parameters which are not shared between BHSPEC and KERRBB. Two\nadditional parameters for BHSPEC are $\\alpha$ and the metal abundance. In\nthe BHSPEC model, the stress is given by\n \\begin{eqnarray} \n\\tau_{R\\phi}=\\alpha P\n\\label{taurphi}\n \\end{eqnarray} \nwhere $\\tau_{R\\phi}$ is the vertically averaged accretion stress and $P$ is the\nvertically averaged total pressure. Usually, the metal abundances are fixed at\nthe solar value, but we also consider fits with three times solar metallicity.\nThe color correction parameter $f$ must be chosen for KERRBB, and we fix\n$f=1.7$ unless stated otherwise. The $f$ value\nwhich brings KERRBB into best agreement with BHSPEC is a function of $\\ell$\n(Shafee et al. 2006), but this choice\nmakes KERRBB roughly consistent over the range of $\\ell$ we consider. We always fix the parameters\n{\\it rflag} and {\\it lflag} so that the spectra are limb darkened and reprocessed\nemission from self-irradiation is ignored. Test cases suggest that these choices\ndo not have a significant effect on the quality of fit or the inferred values for\n$a_{\\ast}$.\n\n\\subsection{Source Selection}\n\\label{soursel}\n\nOur work is motivated in part by spectral fitting of BHBs performed by Gierlinski\n\\& Done (2004). They already provided a sample of sources with {\\it RXTE}\nobservations in which a low fraction (under 15\\%) of the bolometric flux is\ninferred to be in the non-thermal component. We focus on three of these sources:\nLMC X-3, XTE J1550-564 (hereafter J1550), and GRO J1655-40 (hereafter J1655). \nEach one ranges over nearly a decade or more in bolometric luminosity (see Figure\n2 of Gierlinski \\& Done 2004). This makes them particularly well suited for\nconstraining the spectral variation over a range of $\\dot{M}$. The properties\nof these sources are summarized in Table \\ref{tbl1}. All have reasonably precise\nmass estimates, and the distances to LMC X-3 and J1655 both have relatively small\nuncertainties. The distance to J1550 is less well constrained, but we still\ninclude it in our sample because it spans the widest range of luminosities.\n\nOne drawback of {\\it RXTE} is its lack of soft X-ray coverage. The thermal\ncomponents of BHBs typically peak at photon energies near or below 1 keV,\nwhereas the {\\it RXTE} band extends down to only $\\sim3$ keV. \nSince one of our goals is to\ntest the applicability of the underlying accretion disk model, we would like to cover\nas much of the SED as possible. Even if we use other observatories, we face the\ndifficulty that most BHBs lie in or near the Galactic plane and are heavily absorbed by\nthe interstellar medium along the line-of-sight. Therefore, we also examine a\n{\\it BeppoSAX} observation of LMC X-3 which has a low line-of-sight absorption\ncolumn. {\\it\nBeppoSAX} is well-suited for this purpose as it covers a very broad range of\nphoton energies extending down to a tenth of a keV, but lacks the effective area of\n{\\it XMM} or {\\it Chandra} and their corresponding pile-up problems for such\nbright sources.\n\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{f2.eps}\n\\caption{The 66\\%, 90\\%, and 99\\% confidence contours in the $a_{\\ast}$--$\\cos i$\nplane for the best fit BHSPEC model\n($\\alpha=0.01$, $i$ free) to the {\\it BeppoSAX} LMC X-3 data. The \nvertical dashed lines mark uncertainty limits inferred for the binary inclination.\n\\label{fig2}}\n\\end{figure}\n\n\\subsection{LMC X-3}\n\\label{lmc}\n\n\\subsubsection{{\\it BeppoSAX} Data}\n\\label{lmcsax}\n\nThe low energy coverage of {\\it BeppoSAX}\nmakes the fits particularly sensitive to our model for the neutral absorption. \nWe therefore include the line-of-sight absorption column as a free parameter in\nour fits to the {\\it BeppoSAX} data. For our best fit BHSPEC model we find \n$N_H=5.92^{+0.31}_{-0.27} \\times 10^{20}$ cm$^{-2}$. This is\nhigher than values inferred from 21 cm absorption ($3.2 \\times 10^{20}$\ncm$^{-2}$, Nowak et al. 2001) or fits to the neutral oxygen edge in observations\nwith the Reflection Grating Spectrometer on-board {\\it XMM-Newton} ($3.8 \\pm 0.8\n\\times 10^{20}$ cm$^{-2}$, Page et al. 2003), but lower than those found in\nprevious modeling of the {\\it BeppoSAX} data\n($7 \\pm 1 \\times 10^{20}$ $\\rm cm^{-2}$, Haardt et al. 2001).\n\nThe unfolded spectrum of the {\\it BeppoSAX} data is shown in\nFigure \\ref{fig1}. The best fit model for BHSPEC with $i=67^{\\circ}$ and\n$\\alpha=0.01$ which was used to generate the unfolded spectrum \nis also shown. The LMC X-3 spectrum\nis very disk dominated and the PDS provides little constraint due to the low count\nrate. Therefore, we fix the optical depth in the COMPTT model at $\\tau=0.5$,\nproviding a flat (in $\\nu F_{\\nu}$) spectrum typical of thermal dominant state\nobservations.\n\nWe provide a comparison of the three spectral models\nin Table \\ref{tbl2}. With $D$ and $i$ fixed in the KERRBB and BHSPEC, each model has \nthe same number of free parameters: two for the soft\/thermal component, one for the \nnon-thermal component, and one for the intervening absorption column for a\ntotal of four free parameters. DISKBB provides a\nconsiderably poorer fit than either KERRBB or BHSPEC. The relativistically\nbroadened spectra are a much better representation of the soft thermal\nemission than the narrower DISKBB. \n\nThe quality of the DISKBB fit is sensitive to\nthe model of the non-thermal emission. If we treat $\\tau$ as a free parameter,\n$\\tau$ drops and\nthe fit improves slightly ($\\chi^2_\\nu=320\/176$) but remains poor compared with KERRBB and BHSPEC.\nThe COMPTT spectrum steepens as $\\tau$ decreases. This extra flux in the `tail'\nof the thermal component compensates for a decrease in $T_{\\rm in}$ which allows\nDISKBB to better approximate the low energy photons. The fit further improves to\n$\\chi^2_\\nu=264\/176$ if we replace COMPTT with a power law. This provides a slightly\nbetter fit than KERRBB but BHSPEC is still preferred. The best fit model now\nrequires a steep power law component $\\Gamma \\sim 2.8$ which is consistent with\nthe best fit model of Haardt et al. (2001). However, the power law flux now {\\it\nexceeds} the DISKBB flux at low energies. This result is unphysical in a picture\nwhere the soft X-ray emission provides the bulk of the seed photons for the\nnon-thermal component. This is also likely the explanation for why the Haardt et\nal. (2001) fits require a larger neutral hydrogen column. This inconsistency\nwas also pointed out by Yao et al. (2005) who find a self-consistent fit with a\nComptonized multitemperature blackbody model. In contrast, the inclusion of \nnon-thermal emission has little effect on the $\\chi^2$ values for KERRBB and\nBHSPEC. Thus, the {\\it BeppoSAX} data can be completely accounted\nfor by a bare accretion disk spectra as long as relativistic effects on the \nspectra are included.\n\nBHSPEC provides a better fit ($\\Delta \\chi^2=-32$) to the data than KERRBB for an\ninclination $i=67^{\\circ}$, a source distance $D=52$ kpc, and $f=1.7$. Allowing\n$f$ to vary from 1.5-1.9 does not improve the KERRBB quality of fit significantly. \nThe prescription for relativistic effects in the two models are essentially\nidentical, so the differences in the spectral shapes are primarily due to the\ndifferent prescriptions for the disk surface emission. The annuli spectra which\nmake up the BHSPEC model have imprints from\nmetal opacities and may differ from color corrected blackbodies by several\npercent. Additionally, annuli at different radii have local spectra which are best\napproximated by different values of $f$, with $f$ usually being higher for the\nhotter, inner annuli. KERRBB assumes one value of $f$ for the whole\ndisk. Though these discrepancies are not at a level which is significantly\ngreater than the intrinsic uncertainties in the BHSPEC model, it is suggestive\nthat a model which includes atomic physics provides a better fit.\n\n\n\\begin{figure}\n\\includegraphics[width=0.46\\textwidth]{f3.eps}\n\\caption{The 66\\%, 90\\%, and 99\\% confidence contours in the $a_{\\ast}$--$D$ plane\nof the best fit BHSPEC model ($\\alpha=0.01$, $i=67^{\\circ}$, $N$ free) to \nthe {\\it BeppoSAX} LMC X-3 data. Here we have let the normalization vary by 20\\%\nabove and below its nominal value of $N=(10\\,{\\rm kpc}\/52\\,{\\rm kpc})^2=0.0370$.\nThe model normalization $N$ is enumerated on the upper horizontal axis. The \nvertical dashed lines mark uncertainty limits associated the distance estimate\nin Table \\ref{tbl1}\n\\label{fig3}}\n\\end{figure}\n\nDespite these differences in the quality of fit, KERRBB and BHSPEC both give\nvalues of $a_{\\ast}\\sim 0.3$ for $i=67^{\\circ}$. The best fit value for $a_{\\ast}$ for BHSPEC\nis a function of $\\alpha$ with $\\alpha=0.1$ giving a lower $a_{\\ast}$ than\n$\\alpha=0.01$. This is the case in all fits and is most simply understood by\nexamining how changes in the parameters either harden (increase the mean\nphoton energy) or soften (lower the mean photon energy) the spectra. As\n$a_{\\ast}$ increases, the inner radius of the disk decreases. This results in a\nlarger fraction of the gravitational binding energy being released in a smaller\narea of the disk surface. The resulting increase in $T_{\\rm eff}$ in\nthese annuli produces a spectrum with higher average photon energies. Therefore,\nincreasing $a_{\\ast}$ hardens the spectra, even at fixed luminosity.\n\nThe sensitivity of the spectrum to $\\alpha$ is more complex. It \nis strongest at high $\\dot{M}$\nin radiation pressure dominated annuli where the surface density $\\Sigma$ is low.\nA larger $\\alpha$ yields a lower $\\Sigma$, making the disk less effectively optically thick.\nFor a range of $\\dot{M}$, the $\\alpha=0.01$ annuli remain very effectively optically \nthick while the $\\alpha=0.1$ annuli become less effectively optically thick and eventually\neffectively optically thin as $\\dot{M}$ increases. For $\\alpha=0.1$, the densities are\nlower and the temperatures are higher. The photons cannot thermalize as well, causing \nthe spectrum to harden significantly. At lower $\\dot{M}$, both\nmodels have sufficiently large $\\Sigma$ so that spectral formation occurs nearer the disk\nsurface at approximately the same densities and temperatures. The $\\alpha=0.1$ models still \ntend to be slightly less dense in the spectral forming region and are therefore slightly harder,\nbut the differences are significantly smaller than at higher $\\dot{M}$.\nTherefore, the dependence of the best-fit $a_\\ast$ on $\\alpha$ results from an\nincrease in $\\alpha$ from 0.01 to 0.1 hardening the spectrum so that $a_{\\ast}$ must be\nreduced to compensate. \n\nThe best fit $a_{\\ast}$ is also a function of $i$. As can be seen in Table\n\\ref{tbl2}, making $i$ and $D$ free parameters does not significantly improve the\nquality of fit in these cases. However, it does greatly increase the\nuncertainty in the best fit $a_{\\ast}$. This is illustrated in Figure\n\\ref{fig2} where we plot the 66\\%, 90\\%, and 99\\% joint confidence\ncontours for $a_{\\ast}$ and $i$. The strong correlation exists because\nlowering $i$ softens the spectrum so that $a_{\\ast}$\nmust increase to compensate. This is partly because the line-of-sight projection of the \nazimuthal fluid velocity component decreases. This reduces the blueshift and \nbeaming of the emission from the\napproaching side of the disk, and moves the `position' of the high energy tail\nto lower energies. Also, the projected disk area increases which produces a\nlarger flux for the observer at infinity. This needs to be compensated by a\ndecrease in $\\dot{M}$, and therefore $\\ell$. This decrease in $\\ell$\nlowers $T_{\\rm eff}$ and again softens the\nspectrum. Thus, our ability to constrain $a_{\\ast}$ is generally improved by\nprecise, reliable estimates for $i$. The contours in Figure \\ref{fig2}\nsuggest $a_{\\ast} \\simeq 0.55 \\pm 0.2$ at 90\\% confidence. If the binary inclination\nuncertainties are accurate and the inner accretion disk is aligned with the binary orbit,\nthese constraints imply $0.2 \\lesssim a_{\\ast} \\lesssim 0.4$.\n\nWe also examine the variation of $a_{\\ast}$ with the source distance $D$. The \nBHSPEC model normalization $N$ is defined so that $D=10\/\\sqrt{N}$ kpc. \nThe confidence range for the\ndistance to LMC X-3 listed in Table \\ref{tbl1} provides a tight constraint on $N$.\nHowever, $N$ also depends on the absolute flux calibration of the detector so that \nany uncertainty in absolute flux translates into an effective uncertainty for $D$. \nTherefore, we consider fits where $N$ is free to vary by 20\\% from its nominal values of \n$N=(10\\,{\\rm kpc}\/52\\,{\\rm kpc})^2=0.0370$. The 66\\%, 90\\%, and 99\\% confidence \ncontours in the $a_{\\ast}$--$D$ plane are shown in Figure \\ref{fig3}. The best fit\nmodel lies at the upper limit of the allowed range of $N$. The best\nfit $a_{\\ast} \\sim 0.38$ is slightly larger than the range ($a_{\\ast} \\sim 0.25 \\pm 0.05$ )\nconsistent with the distance constraints which are plotted as vertical dashed lines in\nthe figure.\n\nThe strong anticorrelation between $a_{\\ast}$ and $D$ seen in Figure \\ref{fig3} exists\nbecause an increase in $D$ leads to a decrease in the flux expected at the detector\n(i.e. a lower $N$). This must be accounted for by an increase in the luminosity ($\\ell$)\nwhich would shift the spectral peak to higher energies at fixed $a_{\\ast}$. However, \n$a_{\\ast}$ is a free parameter and it can be lowered so that the spectral peak remains \nfixed while $\\ell$ increases.\n\n\n\\begin{deluxetable*}{lccccccc}\n\\tablecolumns{8}\n\\tablecaption{LMC X-3 {\\it BeppoSAX} Fit Summary\\label{tbl2}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Model\\tablenotemark{a}} &\n\\colhead{$\\alpha$} &\n\\colhead{$i$} &\n\\colhead{$D$} &\n\\colhead{$a_{\\ast}$} &\n\\colhead{$k T_{\\rm in}$} &\n\\colhead{$N_H$} &\n\\colhead{$\\chi^2_\\nu$} \\\\\n &\n &\n\\colhead{(deg)} &\n\\colhead{(kpc)} &\n &\n\\colhead{(keV)} &\n\\colhead{$(10^{20}\\,{\\rm cm}^{-2})$} &\n\\\\\n}\n\n\\startdata\nDISKBB &\n\\nodata &\n\\nodata &\n\\nodata &\n\\nodata &\n$1.0139^{+0.0074}_{-0.0080}$ &\n$4.18^{+0.21}_{-0.20}$ &\n336\/177 \\\\\nKERRBB &\n\\nodata &\n67 &\n52 &\n$0.3639^{+0.0013}_{-0.0013}$ &\n\\nodata &\n$5.08^{+0.27}_{-0.25}$ &\n275\/177 \\\\\nBHSPEC &\n0.1 &\n67 &\n52 &\n$0.141^{+0.021}_{-0.020}$ &\n\\nodata &\n$5.65^{+0.27}_{-0.26}$ &\n243\/177 \\\\\nBHSPEC &\n0.01 &\n67 &\n52 &\n$0.258^{+0.019}_{-0.019}$ &\n\\nodata &\n$5.58^{+0.27}_{-0.26}$ &\n246\/177 \\\\\nBHSPEC &\n0.01 &\n$53^{+13}_{-10}$ &\n$51.4^{+1.2}_{-0.0}$ &\n$0.54^{+0.11}_{-0.10}$ &\n\\nodata &\n$5.92^{+0.31}_{-0.27}$ &\n238\/175 \\\\\n\n\\enddata\n\n\\tablenotetext{a}{The full Xspec model is WABS*(Model+COMPTT).}\n\n\\tablecomments{All uncertainties are 90\\% confidence for one parameter.\nParameters reported without uncertainties were held fixed during the fit.}\n\n\\end{deluxetable*}\n\nWe also use these models to test for the possibility of magnetic torques on the\ninner accretion disk. The energy release by a torque increases the fraction of\nemission at small radii and increases the effective temperature of the annuli. \nThis produces a hardening of the spectrum similar to an increase in $a_\\ast$. \nAt the time of publication, BHSPEC spectra have only been computed from disks with \nnon-zero torques for $a_\\ast=0$. KERRBB can be used to examine torques at all\n$a_{\\ast}$, but the fits provide little constraint as $\\Delta \\eta\/\\eta$ varies over\nthe entire range of the model from zero to one at 90\\% confidence when $i$ is a free\nparameter. As expected, increases in $\\Delta \\eta\/\\eta$ are offset by decreases \nin $a_\\ast$. An upper-limit on torque can be obtained by fitting BHSPEC at \n$a_\\ast=0$. The best fit $\\Delta\\eta\/\\eta=3 \\pm 0.8$ with $\\chi^2\/\\nu=245\/175$ \nand $i=73^\\circ \\pm 1^\\circ$. This value of $\\chi^2$ is only slightly greater than\nin the untorqued case, and the best fit inclination is consistent with the \nconstraints on the binary inclination, so it is difficult to rule out the \npossibility of large torques from these data.\n\n\\subsubsection{{\\it RXTE} Data} \n\nGD04 have already\nselected a sample of disk dominated {\\it RXTE} observations for several sources, including\nLMC X-3. From these, we have selected a subset of 10 epochs which evenly cover\nthe range of disk luminosities inferred from the GD04 analysis. The luminosities\nof these epochs are plotted versus the maximum color temperature in the first panel\nof Figure \\ref{fig4}. The black filled circles correspond to the data sets used\nin our work and red triangles represent the other epochs in the GD04 sample. \nThese plots were generated by taking DISKBB fit results and making corrections\nfor the temperature profile and relativistic effects (Zhang et al. 1997). This\nplot only includes epochs in which the disk component is inferred to account for\ngreater than 85\\% of the bolometric flux. A detailed explanation of the analysis\ncan be found in GD04. The values of $L_{\\rm disk}\/L_{\\rm Edd}$ and $T_{\\rm max}$\nare evaluated using the estimates in Table \\ref{tbl1} so there is some\nuncertainty in collective position of these symbols on the plot. However, the\nplacement of the points relative to each other is robust to these uncertainties\nso that reproduction of the shapes of these $L-T$ relations provides an important test\nfor our disk models. We repeat the same procedure for observations of J1550 \nand J1655 and plot these in the center and right panels of Figure \\ref{fig4},\nrespectively.\n\n\nAs stated in section \\ref{intro}, the luminosity is roughly proportional to the\nfourth power of the maximum temperature. The dashed curves represent lines of\nconstant $f$ where $L \\propto T_{\\rm max}^4$. In the bottom panels of Figure\n\\ref{fig4} we have also plotted $L\/T_{\\rm max}^4$ in order to more easily evaluate\nspectral hardening relative to this overall trend. Comparison of the data with these\ncurves shows some evidence for hardening with increasing $L$ for J1665 and J1550. \nLMC X-3 is roughly consistent with a constant $f$, but a close examination suggests \nthere might be weak signs of hardening above $\\sim 0.9$ keV and\nsoftening at the highest temperatures.\n\nWe investigate this spectral evolution with $\\dot{M}$ by fitting the models\ndirectly to the data. We consider the same models as in section \\ref{lmcsax}, but\nwe now fix the absorption column since it is not well constrained without the low\nenergy coverage. For LMC X-3 we fix it at $N_H=5.5 \\times 10^{20}$ $\\rm cm^{-2}$\nto be consistent with the {\\it BeppoSAX} fits.\nWe initially fix $i$ and $D$, and fit only a single value of $a_\\ast$ for all epochs.\nWe also fix $f=1.7$ for KERRBB and $\\alpha=0.1$ or 0.01 for BHSPEC. Only\n$\\dot{M}$ (or $\\ell$) is allowed to vary for each epoch.\nWe also consider models with DISKBB, fitting a single \nnormalization simultaneously to all data sets. This is also consistent with assuming\na fixed color correction and constant effective area for each epoch. With these\nchoices, each model has the same number of free parameters. There is a single parameter \nshared by all data sets ($a_{\\ast}$ or DISKBB normalization), and two parameters\nfor each individual data set: one for the soft\/thermal component ($\\dot{M}$, $\\ell$, or\n$T_{\\rm in}$) and a normalization for the non-thermal component. For BHSPEC, we\nalso consider fits where $i$ and $D$ (or $N$) are free parameters. Since only a\nsingle value of either parameter is fit for all epochs, this provides at most\ntwo additional parameters.\n\nIn oder to visualize the variation of the spectral shape with $\\dot{M}$,\nwe plot in Figure \\ref{fig4} the $L-T$ relations (solid curves) derived from\nthe best fit BHSPEC models at fixed $i$ for $\\alpha$=0.1 and 0.01. These curves are \ncalculated by generating artificial spectra with our best fit models, and then fitting \nthem using the same procedure that GD04 used for the real data. The plot therefore\nprovides a comparison of fits with DISKBB, both to the data, and to artificial spectra\ngenerated from the best-fit BHSPEC models. It is {\\it not} a direct comparison of BHSPEC\nwith the data. This explains\nwhy the best-fit BHSPEC curves do not go through the `data points' in the J1550 and J1655 \nplots. The two types of fits find a different partition of the spectra between the \nsoft\/thermal and hard\/non-thermal components, with additional flux accounted for\nby the non-thermal emission in the BHSPEC fits. The implications of this are discussed\nfurther in section \\ref{hard}. For comparison, we also show curves with \n$L_{\\rm disk}\/L_{\\rm Edd} \\propto T_{\\rm max}^3$ (dotted lines). These curves\nrepresent simple, analytic estimates for the spectral hardening in effectively\noptically thick disks when the effects of Comptonization are negligible. The\nderivation of this relation in presented in the Appendix.\n\n\\begin{figure*}\n\\includegraphics[width=0.94\\textwidth]{f4.eps}\n\\caption{The disk luminosity (top panels) as a function of maximum temperature \nfor J1655, J1550,\nand LMC X-3. Each symbol represents a DISKBB fit to one {\\it RXTE} data set. \nThese data were presented in GD04 and the reader is referred there for a complete\ndiscussion of their spectral analysis. The black filled circles represent the\nepochs which were used in this work and the red triangles represent the remaining\nGD04 data sets. The dashed curves are lines of constant color correction\ncorresponding to $f=1.6, 1.8, 2.0, 2.2, 2.4, 2.6$, and 2.8 for a Schwarzschild\nblack hole (see eq. 3 of GD04). The dotted lines represent curves with\n$L_{\\rm disk}\/L_{\\rm Edd} \\propto T_{\\rm max}^3$. This $L-T$ relation follows\nfrom simple, analytic estimates for the spectral hardening in effectively\noptically thick disks when the effects of Comptonization are negligible. The\nnormalization is chosen arbitrarily to compare with the BHSPEC model curves. The\nderivation of this relation and its relevance to the models is discussed\nin the appendix. The green and blue curves show the evolution \nexpected from the best fit BHSPEC models for $i$ fixed at the binary estimate\n(see Table \\ref{tbl1}) for $\\alpha=0.1$ and 0.01,\nrespectively. These curves were created by producing synthetic data sets with \nthe BHSPEC model and replicating the spectral analysis performed by GD04. \nTo more easily evaluate the spectral hardening relative to a fixed $f$,\nwe have plotted $L\/T_{\\rm max}^4$ (lower panels). The units on the vertical\ncoordinates are arbitrary, but the lines of constant $f$ (now horizontal) are\nretained for reference when comparing with the top panel.\n\\label{fig4}}\n\\end{figure*}\n\n\nBased on the discussion above, it is reasonable to expect that\nthe difference in quality of fit between models with fixed $f$ (KERRBB and DISKBB)\nand BHSPEC might be dominated by the differing predictions for the variation of spectral\nhardening with changing $\\ell$.\nThe lack of hardening at the highest $\\ell$ seen in Figure \\ref{fig4} for LMC X-3 suggests \nthat models with constant $f$ or BHSPEC with a low value of\n$\\alpha$ would provide the best representations of the data. These\npredictions are borne out in simultaneous fits to the LMC X-3 data, which are\nsummarized in Table \\ref{tbl3}. The unfolded spectra are plotted in\nFigure \\ref{fig5} with the best fit BHSPEC model for $\\alpha=0.01$ and\n$i=67^{\\circ}$. DISKBB, which is representative of a disk with a fixed emitting\narea and constant $f$, provides the best fit. A comparable $\\chi^2$ is provided by\nKERRBB with $i=67^{\\circ}$ and $f=1.7$. The fit with BHSPEC for $\\alpha=0.01$\ngives an acceptable $\\chi^2$, but provides a poorer representation than the fixed \n$f$ models. The $\\alpha=0.1$ model does not provide an acceptable fit.\nFor BHSPEC, the largest contributions to $\\chi^2$ come from the highest luminosity\nepoch. As seen in Figure \\ref{fig4}, the BHSPEC models seem to harden too rapidly to\naccommodate all epochs simultaneously. BHSPEC hardens more rapidly with \nincreasing $\\ell$ for $\\alpha=0.1$ than 0.01, leading to the significantly\npoorer fit. In order to make the inner disk annuli more\neffectively thick, we increased the metal abundances to three times the solar\nvalue. However, this had limited impact on the spectral shape and did not\nimprove the quality of fit appreciably.\n\nThe best fit values of $a_{\\ast}$ and their 90\\% confidence intervals are\nsummarized in Table \\ref{tbl3}. They are systematically lower ($a_{\\ast} \\lesssim\n0.1$) than the values inferred from the {\\it BeppoSAX} fits ($a_{\\ast}\\sim 0.3$). \nMost of this discrepancy can be accounted for by cross calibration differences\nbetween the two observatories. Cross-calibration\\footnotemark \\, campaigns on 3C 273\nshow that the {\\it RXTE} PCA flux is about 20\\% higher than the {\\it BeppoSAX} data.\nAt fixed normalization, this requires a $20\\%$ increase in $\\ell$ which leads\nto an $\\sim 5\\%$ increase in $T_{\\rm eff}$. For a given $i$, this change must be offset by \na decrease in $a_\\ast$ which leads to a systematically lower value for the {\\it RXTE}\nfits relative to {\\it BeppoSAX} fits.\n\n\n\\footnotetext{http:\/\/heasarc.gsfc.nasa.gov\/docs\/asca\/calibration\/3c273\\_results.html}\n\n\\subsection{XTE J1550-564}\n\\label{1550}\n\nAs seen in the middle panels of Figure \\ref{fig4}, J1550 varies over an order of\nmagnitude in $\\ell$ and we have chosen 10 observations which sample this range. We\nfit J1550 (and J1655) with the same model which we applied to LMC X-3, but we\nfound statistically significant residuals consistent with reflection features.\n(These were apparently unnecessary in LMC X-3 because the spectra are so strongly\ndisk dominated.)\nTo account for these residuals we add a GAUSSIAN and apply a SMEDGE (Ebisawa et\nal. 1991) to the COMPTT component to approximate reprocessing at the disk\nsurface. We fix the widths of these components to 0.5 keV and 7 keV,\nrespectively. This adds two free parameters from each new component. We also\nlet the optical depth vary in the COMPTT component for a total of\nfive additional parameters in each epoch. The fit results with these additional\ncomponents are summarized in Table \\ref{tbl3}.\n\nAs with LMC X-3, all of the models provide an acceptable fit, though there is\nstill considerable variation in the $\\chi^2$ values. There is a statistically\nsignificant preference for the DISKBB model over KERRBB. Since both models have\na constant $f$, the difference in the quality of fit must be due to the overall\nspectral shape and not simply its evolution with $\\dot{M}$. A comparison of\nthe best fit spectral shapes shows that relativistic KERRBB models are harder\nthan those of DISKBB over the {\\it RXTE} band, which seems to be primarily the\nresult of relativistic\nbroadening.\n\nThe BHSPEC model with $\\alpha=0.01$ provides a better fit than $\\alpha=0.1$,\nconsistent with the prediction of the $L-T$ comparison in Figure \\ref{fig4}. \nHowever, the $\\alpha=0.01$ model still provides a poorer fit than DISKBB. In\ncontrast to KERRBB, the best fit spectra fall off more steeply with increasing photon\nenergy than the DISKBB spectra. This behavior seems to be primarily due to\nabsorption features (primarily Fe K) in the tail of spectrum. When we let $i$\nand $D$ float, the BHSPEC fit improves and the $\\chi^2$ values are now slightly\nbetter than DISKBB. The best fit inclination is lower ($i=42^{+3}_{-13}$ deg)\nand the spin is higher ($a_{\\ast}=0.72^{+0.15}_{-0.01}$),\nproducing slightly harder spectra with less pronounced absorption features than\nin the $i=72^{\\circ}$ case.\nFor $i=72^{\\circ}$, the best fit $a_{\\ast}$ is relatively low ($\\lesssim 0.1$)\nfor both KERRBB and BHSPEC, so allowing $i$ to vary changes $a_{\\ast}$ considerably.\n\nApparent motion $\\gtrsim 2 c$ has been claimed to be observed in the radio\nemission from this source (Hannikainen et al. 2001) suggesting $i \\lesssim\n50^{\\circ}$ for ballistic motion. This implies a misalignment of at least\n$20^{\\circ}$ if the jet is aligned with the black hole angular momentum vector.\nThe best fit inclination $i=42^{+3}_{-13}$ deg is consistent with this upper\nlimit and may be compatible with an inner disk aligned with the black hole via\nthe Bardeen-Petterson effect.\n\n\\subsection{GRO J1655-40}\n\\label{1655}\n\nOf the three sources considered in this work, J1655 displays the strongest\nevidence of hardening in Figure \\ref{fig4}, suggesting that the models with fixed\n$f$ will provide poorer fits than BHSPEC. In fact, KERRBB does not provide an\nadequate fit to the data, and a luminosity dependent trend can be seen in the\nresiduals. However, DISKBB can still provide an adequate fit and is even\npreferred to BHSPEC for $i=70^{\\circ}$. As in J1550, the best fit KERRBB spectra\nare harder than those of DISKBB. These results again suggest that the overall\ndifferences in spectral shape (as opposed to the variation with $\\dot{M}$)\nprovide the dominant effect on the quality of fit. Unlike LMC X-3 and J1550, the\nBHSPEC model with $\\alpha=0.1$ provides a better fit than $\\alpha=0.01$.\nComparison with the bottom-right panel of Figure \\ref{fig4} suggests that the soft, thermal\ncomponent in J1655 is hardening more strongly with increasing luminosity than can\nbe easily accounted for with the $\\alpha=0.01$ model.\n\nAt fixed inclination, BHSPEC with $\\alpha=0.1$ provides the only relativistic fit\nwhich is marginally acceptable. The best fit $a_{\\ast}=0.62 \\pm 0.01$ in this\ncase suggests a moderate spin which is reasonably consistent with other\ninvestigations ($a_{\\ast}\\sim 0.7-0.9$, Gierlinski et al. 2001; $a_{\\ast}\\sim\n0.65-0.75$, Shafee et al. 2006). KERRBB also yields a similar spin \n($a_\\ast=0.6015^{+0.0013}_{-0.0023}$) for $f=1.7$. The small discrepancy with\nthe Shafee et al. (2006) is likely due to our choice of a single $f$ and\nto differences in our modeling of the non-thermal emission.\n\nAllowing $i$ to be a free parameter significantly reduces\n$\\chi^2$ for both values of $\\alpha$, but $\\alpha=0.01$ now provides a slightly\nbetter fit. As with LMC X-3 and J1550, the spin is very sensitive to the inclination.\nFor both values of $\\alpha$ the best fit $a_{\\ast}=0$, the lower limit of the model.\n(We have not yet extended BHSPEC to retrograde spins.) \nWe find $\\chi^2_\\nu=248\/302$ and $i=83.8^{\\circ} \\pm 0.6^{\\circ}$ for\n$\\alpha=0.1$, and $\\chi^2_\\nu=234\/302$ and $i=85.6^{\\circ} \\pm 0.4^{\\circ}$\nfor $\\alpha=0.01$.\n\nRadio observations of J1655 have inferred a jet inclination of $85^{\\circ} \\pm 2^{\\circ}$\nto the line of sight (Hjellming \\& Rupen 1995). Assuming the jets are aligned\nwith the angular momentum axis of the black hole, this would imply an inner disk\ninclination (to the plane of the sky) of $i=85^{\\circ}$, consistent with the best\nfit $i$ above. However, disk alignment only occurs for black holes with\nnon-zero angular momentum and the transition radius is expected to increase with\nincreasing spin (Bardeen \\& Petterson, 1975). Therefore, because these fits find\n$a_\\ast=0$, they do not provide a self-consistent picture for a misaligned disk\nscenario.\n\n\\begin{deluxetable*}{lccccccc}\n\\tablecolumns{8}\n\\tablecaption{{\\it RXTE} Fit Summary \\label{tbl3}}\n\\tablewidth{0pt}\n\\tablehead{\n &\n &\n\\multicolumn{2}{c}{LMC X-3} &\n\\multicolumn{2}{c}{XTE J1550-564} &\n\\multicolumn{2}{c}{GRO J1655-40} \\\\\n\\colhead{Model\\tablenotemark{a}} &\n\\colhead{$\\alpha$} &\n\\colhead{$a_{\\ast}$} &\n\\colhead{$\\chi^2_\\nu$} &\n\\colhead{$a_{\\ast}$} &\n\\colhead{$\\chi^2_\\nu$} &\n\\colhead{$a_{\\ast}$} &\n\\colhead{$\\chi^2_\\nu$} \\\\\n}\n\n\\startdata\nDISKBB\\tablenotemark{b} &\n\\nodata &\n\\nodata &\n296\/431 &\n\\nodata &\n230\/355 &\n\\nodata &\n284\/304 \\\\\nKERRBB\\tablenotemark{c} &\n\\nodata &\n$0.119^{+0.013}_{-0.013}$ &\n296\/431 &\n$0.097^{+0.005}_{-0.065}$ &\n301\/355 &\n$0.6015^{+0.0013}_{-0.0023}$ &\n439\/304 \\\\\nBHSPEC &\n0.1 &\n$0$ &\n1190\/431 &\n$0^{+0.0055}_{-0}$ &\n324\/355 &\n$0.617^{+0.013}_{-0.006}$ &\n330\/304 \\\\\nBHSPEC &\n0.01 &\n$0^{+0.006}_{-0}$ &\n359\/431 &\n$0.115^{+0.030}_{-0.011}$ &\n256\/355 &\n$0.639^{+0.012}_{-0.006}$ &\n369\/304 \\\\\nBHSPEC\\tablenotemark{d} &\n0.01 &\n$0.728^{+0.036}_{-0.018}$ &\n307\/429 &\n$0.72^{+0.15}_{-0.01}$ &\n221\/353 &\n$0.00^{+0.021}_{-0}$ &\n234\/302 \\\\\n\\enddata\n\n\\tablenotetext{a}{The full Xspec model is WABS*(Model+COMPTT) for LMC X-3, and\nWABS*(Model+GAUSSIAN+SMEDGE*COMPTT) for J1550 and J1655.}\n\\tablenotetext{b}{A single normalization is fit for all data sets.}\n\\tablenotetext{c}{The hardening factor is fixed at $f=1.7$. Fit parameters were \nselected so that limb darkening was included but self-irradiation was not.}\n\\tablenotetext{d}{Both $D$ and $i$ are free parameters with the value of $i$ \nallowed to vary over the full range but $D$ constrained to lie within the \nconfidence limits reported in Table \\ref{tbl1}. The best fit values are $\ni=43.4^{+8.5}_{-4.0}$ deg,\n$D=51.4^{+0.61}_{-0}$ kpc for LMC X-3; $i=42^{+3}_{-13}$ deg\n$D=6.3510 \\pm 0.0080$ kpc for J1550; and $i=85.60^{+0.14}_{-0.38}$ deg, \n$D=3.4000^{+0}_{-0.0059}$ kpc for J1655.} \n\n\\tablecomments{All uncertainties are 90\\% confidence for one parameter.\nParameters reported without uncertainties were held fixed during the fit.\nUnless otherwise noted, the values of $i$, $D$, and $M$ were fixed at the \nestimates given in Table \\ref{tbl1}.}\n\n\\end{deluxetable*}\n\n\n\\section{Discussion}\n\\label{discus}\n\nOne of the principle aims of this work was to test the applicability of the\nrelativistic $\\alpha$-disk model in BHBs. The spectral fitting discussed in\nsection \\ref{specmod} presents mixed results. The significant improvements in\n$\\chi^2$ relative to DISKBB resulting from the BHSPEC and KERRBB fits to the {\\it\nBeppoSAX} data provide a strong case for relativistic broadening. DISKBB alone\nis too narrow to adequately approximate the soft, thermal emission from LMC X-3. \nThe additional quality of fit improvement for BHSPEC relative to KERRBB might\nalso be evidence for modified blackbody and smeared absorption features in the\nspectrum.\n\nIn light of these results, it is surprising that DISKBB with a fixed\nnormalization seems to provide a better fit to the {\\it RXTE} data than KERRBB\nor BHSPEC in all three sources when we fix $i$ at the binary inclination. Since\nit is preferred to both relativistic models, the difference cannot simply be due\nto the differences in the degree of spectral hardening as luminosity changes. This could be\ntaken as evidence against relativistic broadening, but the innermost radii\nimplied by the DISKBB fits are consistent with coming from near the black hole. \nA comparison of the best fit spectral shapes for all three models shows\ndifferences at the $\\lesssim 10\\%$ level. The KERRBB spectral shapes tend to be\nbroader (harder in the 3-20 keV band) than both DISKBB and BHSPEC. The\ndifferences between BHSPEC and KERRBB seem to be mostly due to broad absorption\nfeatures which cause the BHSPEC model to fall off more strongly with increasing\nenergy in the tail of the spectrum. Therefore, it is conceivable that DISKBB\nspectrum could be mimicking similar, but slightly weaker, features in data,\nthough it is surprising that it does this consistently and effectively in all\nthree sources.\n\nAlternatively, it may simply be that our estimates for $D$, $M$, or $i$ are in\nerror. When we allow $i$ to be a free parameter, BHSPEC\nprovides a better fit than DISKBB in both J1550 and J1655. In \nboth cases, observations of the radio jets\nsuggest the angular momentum of the black hole is misaligned with that of the\nbinary. It is suggestive that in both cases we find values for $i$ consistent\nwith the constraints implied by the jets, rather than the binary inclination.\nIn the case of LMC X-3, where a jet has not been observed, the best fit $i$\nis more nearly face on than the measured binary inclination. It is consistent\nwith the binary inclination at 90\\% confidence for the {\\it BeppoSAX} data, but\nnot for the {\\it RXTE} spectral fits.\n\nA third possibility is that the non-thermal emission and Compton reflection \ncomponents are not being correctly accounted for by our prescription. If\nthis is a problem, it should be minimized by looking at LMC X-3, which has\nthe most disk dominated spectra of the three sources and relatively little\nevidence for reflected emission.\nFor LMC X-3 the BHSPEC residuals are clearly dominated by the most luminous epoch\nfor which BHSPEC predicts too much hardening with increasing luminosity for\neither value of $\\alpha$. If we ignore the most luminous epoch $\\chi^2_\\nu$\nimproves to 257\/388 for BHSPEC with $\\alpha=0.01$ and $i=67^{\\circ}$, but only\nimproves slightly to 256\/388 for DISKBB, providing comparable fits. Allowing $i$ \nto be a free parameter allows the BHSPEC fit to improve even further for slightly\nmore face on values. If $i$ remains fixed at $67^{\\circ}$, the fit with BHSPEC also\nimproves by allowing the model normalization to vary within 20\\% of the nominal\nvalue. This is a larger range of normalization than that associated with\ndistance uncertainty and accounts for possible errors in the absolute flux\ncalibration. Thus for LMC X-3, the shape of the spectra seem to be\nbest represented by BHSPEC, but the evolution of the spectral hardening with\n$\\ell$ at the highest luminosities is not consistent with the predictions of a\nsimple $\\alpha$-disk model.\n\n\\subsection{Spectral Hardening and the Stress Prescription}\n\nThe three $L-T$ relations presented in Figure \\ref{fig4} show that differences\nexist in the spectral evolution with disk luminosity from source to source. LMC\nX-3 has the most scatter and is reasonably consistent with a constant $f$, \nthough the lower-left panel of Figure \\ref{fig4} shows evidence of a weak\nhardening for $0.1 \\lesssim \\ell \\lesssim 0.3$ and softening for $\\ell \\gtrsim 0.3$.\nJ1550 is consistent with weak hardening and J1655 seems to show more significant\nhardening. A comparison of the KERRBB and BHSPEC fit results seems to agree with\nthese descriptions. At fixed $i$, KERRBB with $f=1.7$ is preferred for LMC X-3, but\nBHSPEC provides a better fit in both J1550 and J1655. Fitting a single $f$ for\nall epochs with KERRBB does not alter this result. A comparison of BHSPEC fits\nwith $\\alpha=0.1$ and 0.01 is also generally consistent. With $i$ fixed at the\nestimate for the binary inclination, $\\alpha=0.01$ is preferred for LMC X-3 and\nJ1550, but $\\alpha=0.1$ provides a better fit for J1655. If $i$ is a free\nparameter, $\\alpha=0.01$ provides a better fit for all three sources, though the\nimprovement is small for J1655.\n\nAs discussed in section \\ref{lmcsax} and the Appendix, the sensitivity of the \nspectra to $\\alpha$ comes about primarily because $\\alpha$ determines $\\Sigma$.\nFor sufficiently large $\\Sigma$, the spectral shape depends only weakly on $\\alpha$.\nHowever, if $\\Sigma$ drops sufficiently, the disks begin to become effectively optically \nthin at small radii. Once the hottest annuli are effectively optically thin, they become \nincreasingly isothermal or inverted in their temperature profiles as inverse Compton\nscattering in the now hotter surface layers increasingly dominates the cooling.\nThis is much less efficient than thermal cooling and the spectra harden rapidly as\ntemperatures rise with increasing $T_{\\rm eff}$ (Davis \\& Hubeny 2006). Such \neffects are responsible for the hardening in the $\\alpha=0.1$ models at high \n$\\ell$ in Figure \\ref{fig4}. Our results\nsuggest that these multi-epoch fits are sensitive to these effects and may\neven be able to differentiate between the $\\alpha$-disk prescription\nand more general models of angular momentum transport in these disks.\n\nIn the context of the $\\alpha$ prescription, our fit results seem to rule out\nfixed values of $\\alpha \\ge 0.1$ for LMC X-3 and possibly J1550.\nAt first sight, this appears inconsistent with the $\\alpha \\ge 0.1$ values\ninferred for the outburst phases of dwarf novae (Lasota 2001) and soft\nX-ray transients (e.g. Dubus et al. 2001). However, it is important to note\nthat the disk instabilities that drive the outburst time scales are\nassociated with regions of the disk where gas pressure dominates radiation\npressure. In contrast, the X-ray spectra are dominated by the innermost\nregions of the disk where radiation pressure can be important at high\n$\\dot{M}$. There is no reason to believe that either $\\alpha$ or the\nstress prescription should be the same at all radii in the disk.\n\nIt may be that the classical stress prescription of equation (\\ref{taurphi})\nis not valid in radiation pressure dominated disks. Alternative stress\nprescriptions have long been considered, partly\nbecause they can produce thermally and viscously stable disks (e.g. Piran\n1978).\nIn addition, there have been proposals that magnetohydrodynamical\nturbulence may produce stresses that are limited to values that are related\nin some way to the gas pressure (e.g. Sakimoto \\& Coroniti 1981, Merloni 2003).\n\nIt is noteworthy that\nLMC X-3 reaches the highest Eddington ratios among the sources fit here, and that\nthe spectrum even appears to {\\it soften} slightly at these highest luminosities.\nA substantial reduction in stress at $\\sim 0.6-0.7$ might explain this, perhaps linked\nto the onset of a disk instability, which appears to be present in the only other\nsource to consistently exceed such luminosities, GRS 1915+105. Fits with BHSPEC\nto GRS 1915+105 show that this source has `stable' disk spectra (i.e. constant\nfor longer than 16 sec) from $\\ell=0.5-0.6$ and from $\\ell=1-2$ which are consistent with\nthe expected hardening with luminosity (Middleton et al. 2006).\nHowever, there are {\\it no} stable disk spectra from this source in the range\n$\\ell \\sim 0.7-0.9$, exactly where the LMC X-3 spectra show slightly different\nproperties than expected.\n\nOther effects might be important at the high luminosities.\nThe disk models used in BHSPEC are actually somewhat inconsistent for\n$\\ell \\gtrsim0.3$, as the innermost annuli then have\n$H\/R\\gtrsim0.1$. Radial transport of accretion power is increasingly\nimportant in this regime. In addition, magnetic torques across the ISCO\nmay be more important (Afshordi \\& Paczy\\'nski 2003). However,\nboth of these effects would tend to make the inner annuli hotter and\/or\nmore effectively optically thin. We would expect this to {\\it increase}\nthe hardening, in contrast to what is observed. As the Eddington ratio\nincreases, increased inhomogeneities in the magnetorotational turbulence \n(Turner et al. 2002) and photon bubbles (Begelman 2001) may mitigate this by producing a\nsofter spectrum and a geometrically thinner disk than would be expected in a homogeneous model.\n\n\n\\begin{figure}\n\\includegraphics[width=0.46\\textwidth]{f5.eps}\n\\caption{The unfolded spectra for the {\\it RXTE}\nobservations of LMC X-3 using the best fit BHSPEC models for $i=67^{\\circ}$, \n$D=52$ kpc,and $\\alpha=0.01$. The total model component (green, solid curve),\nBHSPEC (red, long-dashed curve) and COMPTT (violet, short-dashed curve) are\nplotted.\n\\label{fig5}}\n\\end{figure}\n\n\\subsection{Estimates for Black Hole Spins}\n\nThe total mass accreted over the lifetime of these sources expected to be\nsmall (King \\& Kolb 1999; Shafee et al. 2006). As a result, only a\nsmall increase in the angular momentum of the black hole is expected, and so our \nspin measurements are likely probing the natal spin \ndistribution of the binaries. Therefore, these low to moderate spin estimates\n($a_\\ast \\lesssim 0.8$) place constraints on black hole formation scenarios. They\nmay also constrain spin-dependent models for jet production (Middleton et al. 2006)\nsince both J1550 and J1655 are microquasars. We characterize these spins\nas `moderate' because even for $a_{\\ast}\\sim 0.8$,\nthe proximity of the ISCO to the event horizon\nand the resulting radiative efficiency ($R\\sim 3 R_g$, $\\eta\\simeq 0.12$) are \nsubstantially less extreme than in the maximally spinning spacetime \n($a_{\\ast}\\sim 0.998$, $R\\sim 1.24 R_g$, $\\eta\\simeq 0.32$). These moderate spins\nare in contrast to estimates for spin at or near $a_\\ast=0.998$ that have been\ninferred from other methods, including spectral fits to the Fe K$\\alpha$ line and \nsome interpretations of the high frequency quasi-periodic oscillations (QPOs).\n\nBroad Fe K$\\alpha$ line emission has been seen in {\\it ASCA} observations of\nJ1550 and J1655 (Miller et al. 2004). The emission was modeled with a\nrelativistic disk line profile calculated in a maximally spinning Kerr spacetime\n(LAOR in Xspec, Laor 1991), but with inner radius allowed to vary. The best fit\ninner radius for J1655 was small with $R_{\\rm in} \\lesssim 2 R_g$, suggesting\n$a_\\ast > 0.9$ and possibly near maximal ($a_\\ast \\sim 0.998$). J1550 is less \nwell constrained with $R_{\\rm in} \\lesssim 4-6 R_g$ depending on the model, though\nit may also be consistent with near maximal spin.\nThe strongest constraints on $a_\\ast$\ncome from the presence of a broad, asymmetric red wing which extends down to\n$\\sim$4 keV in the best-fit models. Therefore, a principle source of\nuncertainty for applying this method is modeling the underlying continuum to\naccurately gauge the shape and extent of the line wing. We refer the reader to\nDone \\& Gierlinski, (2005) for a more detailed discussion of uncertainties associated\nwith this method.\n\nThe reproducibility of the frequencies of the pair of high frequency QPOs from one\nobservation to the next suggests they might also provide a direct probe of the\nblack hole spacetime. As a result, prospective models of QPOs often provide\nconstraints on $a_\\ast$. For example, if the lower frequency member of the pair\nin J1550 and J1655 is identified as an\naxisymmetric radial epicyclic oscillation, then black hole spins of $a_\\ast >\n0.9$ are required (Rezzolla et al. 2003, T\\\"or\\\"ok 2005). The reason is simple:\nthe radial epicyclic frequency has a maximum at some radius, and that maximum is\nbelow the observed QPO frequency for the observed black hole masses unless the\nspin is high. The same conclusion holds in diskoseismology models if the lower\nfrequency member is identified with a low order axisymmetric ``g-mode'', because\nit has a frequency less than the radial epicyclic frequency within the mode\ntrapping region (e.g. Wagoner et al. 2001). However, there are many possible\noscillation modes in accretion disks, and other mode identifications can be made\nwhich are more consistent with moderate spins (e.g. Blaes et al 2006a).\n\nIn principle, the high signal-to-noise {\\it RXTE} data allow us to place\nvery tight constraints on the spins of BHBs (see e.g. Table 3). However, the\nrelatively small uncertainties in Table 3 do not account for uncertainties in\n$M$, $i$, and $D$. There are degeneracies among the fitting parameters in the way\nthat they affect the spectrum, e.g. the correlation\nbetween $i$ and $a_\\ast$ seen in Figure \\ref{fig2}. Therefore, when $i$ and $D$ are\nfree parameters, $a_\\ast$ can \nchange significantly. For the {\\it RXTE} data, the uncertainty ranges for \n$a_\\ast$ increase when $i$ and $D$ are free, but still remain relatively small.\nThis also leads to rather small formal uncertainties for $i$ and $D$, though \n$D$ is at the limit of allowed range for LMC X-3 and J1655.\nSo these data are capable of constraining all the parameters simultaneously, \nbecause changes of only a few percent in the spectral shape in the high energy\ntail of the spectrum lead to substantial \nchanges in $\\chi^2$. However, we caution that the models themselves are uncertain\nat the few percent level due to our interpolation method alone (Davis \\& Hubeny 2006).\nTherefore, this method of spin estimation can\nonly be used with good precision when reliable and precise constraints on \n$M$, $i$, and $D$ are available.\n\nAll of these methods of spin estimation have their weaknesses, as all rely to \nvarying degrees on uncertain physical assumptions. An important example relevant\nfor both Fe K$\\alpha$ line and our continuum spectral fits is the assumption\nthat the disk emission extends to the ISCO, and effectively ceases interior to\nthis radius. In principle, the disk (or its emission) could be truncated at larger\nradius and our spins would be underestimates. Alternatively, significant\nemission might be generated or reprocessed inside the ISCO (Krolik \\& Hawley 2002), \nmaking the interpretation of both estimation methods more difficult.\n\nSeveral uncertainties also remain in other physical assumptions which underly\nthe BHSPEC model, because we still lack a complete\nunderstanding of the magnetohydrodynamical structure of the accretion flows.\nThere are several modifications which likely lead to a hardening of the spectra,\nincluding\nincreased dissipation near the disk surface (Davis et al. 2005), an increase\nin the density scale height due to magnetic pressure support (Blaes et al. 2006b), \nsurface irradiation by the non-thermal emission, and torques on the\ninner edge of the disk. Other processes might lead to a softening of the spectra.\nOpacity due to bound-bound transitions of metal ions, which is not included in\nBHSPEC should increase\nthe ratio of absorption to electron scattering opacity, pushing the\nspectrum closer to blackbody. Inhomogeneities, such as those caused by \ncompressible magnetohydrodynamical turbulence (Turner et al. 2002) or the photon bubble\ninstability (Turner et al. 2005), may also soften the spectrum. They may increase the\neffective ratio of \nabsorption to scattering opacity because photon-matter interactions are\ndominated by the densest regions (Davis et al. 2004), or through a reduction \nin the density scale height (see e.g. Begelman 2001) leading to an increase in\nthe average density of the disk interior. Given all these possibilities, it is \ndifficult to say with certainty what net effect modifying our assumptions or\nincluding these additional processes would have on the disk spectra. If the effects\nwhich harden the spectra are more important, BHSPEC \nwill underestimate the spectral hardening and would then require \nhigher spins to fit the data, making our spin measurements overestimates\nand vice-versa.\n\nGiven these uncertainties, it is conceivable that one could reconcile the\nspectral constraints with $a_\\ast \\sim 0.9$ if the BHSPEC model overestimates the\nactual spectral hardening. Spins this high would bring our estimates in\nline with the published uncertainties for the Fe K$\\alpha$ estimates and some\nof the QPO-based measurements. Reconciling our spectral models\nwith the extreme spin ($a_\\ast \\sim 0.998$) is more difficult. We have\nattempted to quantify this in the case of J1655 by fitting the data with KERRBB for\n$a_\\ast=0.998$, but with $f$ as a free parameter. For $i=70^{\\circ}$ or $85^{\\circ}$, \nan adequate fit can be obtained only for $f \\sim 1$ which corresponds to blackbody \nradiation. Electron scattering opacity dominates at the relevant temperatures so \nnearly blackbody emission is highly unlikely. One could obtain agreement\nwith more reasonable color corrections ($f \\sim 1.5$)\nby allowing $i$ to be a free parameter. However, this requires $i\\lesssim 40^{\\circ}$,\nin disagreement with the inclinations from both the binary and jet observations.\n\n\n\\subsection{Uncertainties due to Hard X-ray Emission}\n\\label{hard}\n\nA potential difficulty for deriving constraints on $\\alpha$ or $a_\\ast$ by\nthis method is the\nneed to account for the non-thermal emission. This is particularly true with {\\it RXTE}\ndata when only the high energy tail of the spectrum is typically observed. \nEven though we infer the models' bolometric\nfluxes to be dominated by the softer thermal component, the\nnon-thermal component accounts for a substantial fraction of the 3-20 keV\nemission in many of the cases. The decomposition of thermal and non-thermal\nemission is clearly dependent on the choice of model for the thermal emission. \nIt can be seen in Figure \\ref{fig4} that the $T_{\\rm max}$ values derived from the\nbest-fit BHSPEC models for J1550 and J1655 are softer (lower $T_{\\rm max}$) at\nfixed $L_{\\rm disk}\/L_{\\rm Edd}$ than those derived by GD04 who fit DISKBB directly \nto the data. This means\nthat emission which was being accounted for by the DISKBB model is partly being\naccounted for by the non-thermal emission in the BHSPEC fits. The fits also\ndepends on the choice of model for the non-thermal component. The COMPTT model\nassumes a single Wien spectrum for the soft photon input. A model with a\nmultitemperature disk spectrum for the seed photon input (THCOMP, Zdziarski et al. 1996)\nprovides more low energy photons for a given temperature. Therefore, a THCOMP\nspectrum which matches COMPTT at higher photon energies will tend to have more\nflux at lower energies. We find that the quality of fit can change significantly\n(e.g. DISKBB is no longer preferred to BHSPEC for J1550 at fixed $i$) if we\nreplace COMPTT with THCOMP, though the best-fit spins and preferred $\\alpha$ values\nseem more robust. The LMC X-3 observations are much more disk dominated \nthan those of J1550 and J1655, and are much less sensitive\nto the choice of model for the non-thermal emission.\n\nThe effects of the non-thermal emission can be minimized if we\nobserve these sources with detectors sensitive to lower photon energies, as most\nof the spectra peak below the lower limit of the {\\it RXTE}\nband. Thus, the {\\it RXTE} fits can be sensitive to changes of only a few percent\nin both the non-thermal emission model and the shape of the high energy tail of \nthe BHSPEC models. The shape of the high energy tail of the BHSPEC spectra is uncertain at\nthe several percent level due to our interpolation scheme (Davis \\& Hubeny 2006).\nLine-of-sight absorption at softer-photon energies will\nput practical limits on such modeling, and makes sources located\nout of the Galactic plane (such as LMC X-3) particularly well suited for this\ntype of study.\n\n\\section{Conclusions}\n\\label{conc}\n\nWe analyze disk-dominated spectra of three black hole binaries: LMC X-3,\nXTE J1550-564, and GRO J1655-40, fitting them with a simple multitemperature\nblackbody model (DISKBB), as well as sophisticated relativistic disk models\n(KERRBB and BHSPEC). For LMC X-3, this includes {\\it BeppoSAX} data, which\ncover the 0.1-10 keV energy band in which the majority of the bolometric\nflux of the disk is emitted. In this case we find a statistically significant\npreference for the relativistic models over DISKBB. At lower significance,\nwe also find a preference for BHSPEC over KERRBB which may suggest the\nspectra are sensitive to atomic and radiative transfer physics which are calculated\nexplicitly in BHSPEC.\n\nWe also examine {\\it RXTE} spectra for each of the three BHBs using simultaneous,\nmulti-epoch fits to each source. When we fix the relativistic model at the \nindependent estimates for the binary inclination in Table \\ref{tbl1}, we find DISKBB\nis preferred over both relativistic models for both J1550 and J1655, though $\\chi^2_\\nu$ is\nstill acceptable in most of the relativistic model fits. If we allow the inclination \nto be a free parameter, BHSPEC is the best-fit model for both sources. The best fit \ninclinations are both consistent with constraints inferred from radio observations\nof the jets in these sources which might be accounted for by a misalignment of \nthe black hole spin with the binary orbital angular momentum. BHSPEC is also\nthe best fit model for LMC X-3, if we ignore the highest luminosity epoch\nwhere additional physics appears to be important. The best-fit inclination is marginally\nconsistent with the constraints on the binary inclination in this source.\n\nUsing the binary inclination estimates in Table \\ref{tbl1}, we are able to derive\nprecise estimates for the black hole spin. However, the inferred values of the spin \nare functions of inclination, and the spin changes if the inclination of the X-ray\nemitting, inner disk annuli differs significantly from independent estimates.\nThe best fit spin is also sensitive to the\nsource distance and absolute flux calibration through the model normalization.\nAccurate and precise estimates for all these parameters as well as the black\nhole mass are therefore a prerequisite for accurate and precise spin estimates.\nWe find relatively moderate spins ($a_\\ast \\lesssim 0.8$), even when inclination\nis a free parameter in our fits. For J1655, our maximum spin estimate is only slightly \nlower than the limits ($a_{\\ast} \\gtrsim 0.9$ at 90\\% confidence) implied by fits to \nFe K$\\alpha$ lines \n(Miller et al. 2004) and certain models of high frequency QPOs (\ne.g. Wagoner et al. 2001; Rezzolla et al. 2003; T\\\"or\\\"ok 2005). The spin of J1550\nis more weakly constrained by the Fe K$\\alpha$ fits (Miller et al. 2004), and is \nconsistent with our estimates. However, for both sources the best fit Fe K$\\alpha$\nmodels are also consistent with nearly `maximal' spins ($a_{\\ast}\\sim 0.998$) which \nwould be difficult to reconcile if the BHSPEC models provide an accurate approximation\nto the spectra of the accretion flows in these sources.\n\nWe also find that our fits are sensitive to the assumed form of the angular momentum \ntransport through its effects on the disk surface density. We consider an $\\alpha$ stress \nprescription with $\\alpha=0.1$ and 0.01, finding approximate qualitative\nagreement between the $L-T$ diagrams and the model predictions (see figure\n\\ref{fig4}). We find that all three BHBs are consistent with a single value\nof $\\alpha$, preferring $\\alpha=0.01$ to $\\alpha=0.1$\n(though only weakly in the case of J1655). These results are in contrast to the \nstandard disk instability model of soft X-ray transients (see e.g. Dubus et al. 2001)\nwhich requires $\\alpha \\gtrsim 0.1$ in the outer disk.\nIf we include the most luminous epoch in our fits to the \n{\\it RXTE} data of LMC X-3, models with constant $f$ provide a better\nfit than BHSPEC with either $\\alpha$. BHSPEC fails to account for the most\nluminous epoch because it predicts continued\nspectral hardening at the highest Eddington ratios while the observed spectra appear \nto {\\it soften}. This suggests the onset of additional physics which \nsoftens the spectrum as the Eddington ratio nears unity.\n\n\\acknowledgements{We gratefully acknowledge Ivan Hubeny for his instrumental contributions\nto the development of BHSPEC. We thank Eric Agol for useful discussions,\nand for making KERRTRANS publicly available. We also thank Marek Gierlinski,\nJulian Krolik, Ramesh Narayan, Rebecca Shafee, and Aristotle Socrates for useful\ndiscussions and\/or assistance. We are grateful for comments on the manuscript from the \nanonymous referee and Jeff McClintock which lead to significant improvements. Part of this\nwork was completed while the authors were hosted by the Kavli Institute for Theoretical \nPhysics at UCSB. This work was supported by NASA grant NAG5-13228, and by the National Science \nFoundation under grant PHY99-07949.\n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\input{sections\/Introduction}\n\n\n\\section{Related Work}\n\\input{sections\/Related_Work}\n\n\n\\section{\\textsc{DProQ} Pipeline}\n\\input{sections\/DProQ_Pipeline}\n\n\n\\section{Experiments}\n\\input{sections\/Experiments}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{sections\/Conclusion}\n\n\n\\section*{Acknowledgments}\nThe project is partially supported by two NSF grants (DBI 1759934 and IIS 1763246), one NIH grant (GM093123), three DOE grants (DE-SC0020400, DE-AR0001213, and DE-SC0021303), and the computing allocation on the Summit compute cluster provided by Oak Ridge Leadership Computing Facility (Contract No. DE-AC05-00OR22725).\n\n\n\n{\n\\small\n\\printbibliography\n}\n\n\n\n\n\n\\subsection{Data}\n\\input{sections\/Datasets}\n\n\\subsection{Evaluation Setup}\n\\textbf{Baselines.} We compare our method with the state-of-the-art method GNN\\_DOVE, an atom-level graph attention-based method for protein complex structure evaluation. Uniquely, it extracts protein interface areas to build its input graphs. Further, the chemical properties of its atoms as well as inter-atom distances are treated as initial node and edge features, respectively. We note that GNN\\_DOVE is the sole baseline we include in this study, as for all other DL-based methods for structural QA, we were unable to locate reproducible training and inference code for such methods. Furthermore, since GNN\\_DOVE has previously been evaluated against several other classical machine learning methods for structural QA and has demonstrated strong performance against such methods, we argue that comparing new DL methods' performance to GNN\\_DOVE can serve as a strong indicator of the DL state-of-the-art for the field.\n\n\\textbf{\\textsc{DProQ} Models.} Besides including results for the standard \\textsc{DProQ} model as well as for GNN\\_DOVE, we also report results on the HAF2 and DBM55-AF2 datasets for a selection of \\textsc{DProQ} variants curated in this study. The \\textsc{DProQ} variants we chose to investigate in this work include \\textsc{DProQ\\_GT} which employs the original Graph Transformer architecture \\cite{dwivedi2020generalization}; \\textsc{DProQ\\_GTE} which employs the GGT with only its edge gate enabled; and \\textsc{DProQ\\_GTN} which employs the GGT with only its node gate enabled.\\par\n\n\\textbf{Evaluation metrics.} We evaluate structural QA performance according to two main criteria. Our first criterion is general ranking ability which measures how many qualified decoys are found within a model's predicted Top-$N$ structure ranking. Within this framework, a model's hit rate is defined as the fraction of protein complex targets for which the model, within each of its Top-$N$ ranks, ranked at least one Acceptable or higher-quality decoy based on the CAPRI scoring criteria \\cite{lensink2014score_set}. In this work, we report results based on models' Top-10 hit rates. A hit rate is represented by three numbers separated by the character \/. These three numbers, in order, represent how many decoys with Acceptable or higher-quality, Medium or higher-quality, and High quality were among the Top-N ranked decoys. Our second criterion with which to score models is their Top-$1$ selected model ranking ability. Within this context, we calculate the ranking loss for each method. Here, per-target ranking loss is defined as the difference between the DockQ score of a target's native structure and the DockQ score of the top decoy predicted by each ranking method. As such, a lower ranking loss indicates a stronger model for ranking pools of structural decoys for downstream tasks.\n\n\\textbf{Implementation Details.} We train all our models using the AdamW optimizer \\cite{loshchilov2017decoupled} and perform early stopping with a patience of $15$ epochs. All remaining hyperparameters and graph features we used are described further in Appendix \\ref{sec:appendix_c}. Source code to reproduce our results and perform fast structure quality assessment using our model weights can be found at \\href{https:\/\/github.com\/BioinfoMachineLearning\/DProQ}{\\texttt{https:\/\/github.com\/BioinfoMachineLearning\/DProQ}}.\n\n\\subsection{Results}\n\\textbf{Blind Structure Quality Assessment on the HAF2 Dataset.} Table \\ref{tab:haf2_hits} reports the hit rate performances of \\textsc{DProQ}, \\textsc{DProQ} variants, and GNN\\_DOVE, respectively, on the HAF2 Dataset. On $13$ HAF2 targets, \\textsc{DProQ} hits (i.e., identifies at least one suitable decoy structure for) $10$ targets on the Acceptable quality level, $9$ targets on the Medium quality level, and $4$ targets on High quality level. For targets with High quality structures, \\textsc{DProQ} and its variants successfully hit all of them. On the HAF2 dataset, GNN\\_DOVE gets a $8\/7\/3$ hit rate on Acceptable, Medium, and High quality targets respectively. However, \\textsc{DProQ} outperforms GNN\\_DOVE on all targets except \\textsc{7NKZ}. On this target, GNN\\_DOVE achieves a better hit rate on High quality decoy structures. Overall, \\textsc{DProQ} shows a better performance than \\textsc{DProQ\\_GT}, \\textsc{DProQ\\_GTE}, and \\textsc{DProQ\\_GTN}. Additionally, for target \\textsc{7MRW}, only \\textsc{DProQ} successfully hits the $5$ Acceptable quality targets and $4$ Medium quality targets, whereas all other methods failed to hit any qualified decoy structures for this protein target.\n\nTable \\ref{tab:haf2_ranking} reports methods' ranking loss performance on each HAF2 target. \\textsc{DProQ} and its $3$ variants outperform GNN\\_DOVE which displays the highest average ranking loss. In particular, \\textsc{DProQ} achieves the lowest average ranking loss of $0.195$ compared to all other methods. Here, \\textsc{DProQ}'s loss is $43\\%$ lower than GNN\\_DOVE's $0.343$ average ranking loss. Moreover, \\textsc{DProQ} achieves the lowest loss on $5$ targets, namely \\textsc{7AWV}, \\textsc{7OEL}, \\textsc{7NRW}, \\textsc{7NKZ} and \\textsc{7O27}. Notably, \\textsc{DProQ\\_GT}, \\textsc{DProQ\\_GTE}, and \\textsc{DProQ\\_GTN}'s ranking losses here are $29\\%$, $34\\%$, and $30\\%$ lower, respectively, than GNN\\_DOVE's average ranking loss.\n\n\\begin{table}\n\\caption{Hit rate performance on the HAF2 dataset. The \\textsc{Best} column represents each target's best-possible Top-$10$ result. The \\textsc{Summary} row lists the results when all targets are taken into consideration.}\n\\label{tab:haf2_hits}\n\\centering\n\\begin{tabular}{lllllll}\n\\toprule\nID & \\textsc{DProQ} & \\textsc{DProQ}\\_GT & \\textsc{DProQ}\\_GTE & \\textsc{DProQ}\\_GTN & GNN\\_DOVE & \\textsc{BEST} \\\\ \n\\midrule\n7AOH & 10\/10\/10 & 10\/10\/10 & 10\/10\/10 & 10\/10\/10 & 9\/9\/0 & 10\/10\/10 \\\\\n7D7F & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 5\/0\/0 \\\\ \n7AMV & 10\/10\/10 & 10\/10\/10 & 10\/10\/10 & 10\/10\/10 & 10\/10\/6 & 10\/10\/10 \\\\ \n7OEL & 10\/10\/0 & 10\/9\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 \\\\ \n7O28 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 \\\\ \n7ALA & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 1\/0\/0 \\\\ \n7MRW & 5\/4\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 10\/10\/0 \\\\ \n7OZN & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 10\/2\/0 \\\\ \n7D3Y & 2\/0\/0 & 5\/0\/0 & 6\/0\/0 & 8\/0\/0 & 0\/0\/0 & 10\/0\/0 \\\\ \n7NKZ & 10\/10\/2 & 10\/10\/1 & 10\/10\/1 & 10\/010\/4 & 10\/9\/9 & 10\/10\/10 \\\\ \n7LXT & 1\/1\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 1\/0\/0 & 10\/10\/0 \\\\ \n7KBR & 10\/10\/10 & 10\/10\/10 & 10\/10\/10 & 10\/10\/10 & 10\/10\/9 & 10\/10\/10 \\\\ \n7O27 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/4\/0 & 10\/10\/0 \\\\ \n\\midrule\n\\textsc{Summary} & \\textbf{10\/9\/4} & 8\/7\/4 & 8\/7\/4 & 8\/7\/4 & 8\/7\/3 & 13\/10\/4 \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Ranking loss performance on the HAF2 dataset. The \\textsc{BEST} row represents the mean and standard deviation of the ranking losses for all targets.}\n\\label{tab:haf2_ranking}\n\\centering\n\\begin{tabular}{llllll}\n\\toprule\nTarget & \\textsc{DProQ} & DProQ\\_GT & \\textsc{DProQ}\\_GTE & \\textsc{DProQ}\\_GTN & GNN\\_DOVE \\\\\n\\midrule\n7AOH & 0.066 & 0.026 & 0.026 & 0.058 & 0.928 \\\\\n7D7F & 0.471 & 0.471 & 0.47 & 0.471 & 0.003 \\\\\n7AMV & 0.01 & 0.021 & 0.017 & 0.019 & 0.342 \\\\\n7OEL & 0.062 & 0.063 & 0.135 & 0.135 & 0.21 \\\\\n7O28 & 0.029 & 0.021 & 0.027 & 0.034 & 0.244 \\\\\n7ALA & 0.232 & 0.226 & 0.226 & 0.226 & 0.226 \\\\\n7MRW & 0.085 & 0.603 & 0.555 & 0.555 & 0.598 \\\\\n7OZN & 0.409 & 0.409 & 0.49 & 0.281 & 0.457 \\\\\n7D3Y & 0.326 & 0.33 & 0.012 & 0.326 & 0.295 \\\\\n7NKZ & 0.164 & 0.175 & 0.175 & 0.164 & 0.459 \\\\\n7LXT & 0.586 & 0.586 & 0.586 & 0.586 & 0.295 \\\\\n7KBR & 0.068 & 0.152 & 0.152 & 0.17 & 0.068 \\\\\n7O27 & 0.03 & 0.079 & 0.079 & 0.079 & 0.334 \\\\\n\\midrule\n\\textsc{BEST} & \\textbf{0.195 \u00b1 0.185} & 0.243 \u00b1 0.206 & 0.227 \u00b1 0.21 & 0.239 \u00b1 0.187 & 0.343 \u00b1 0.228 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\textbf{Blind Structure Quality Assessment on the DBM55-AF2 Dataset.} Table \\ref{tab:dbm_hits} summarizes all methods' hit rate results on the DBM55-AF2 dataset, which contains $15$ targets. Here, we see that \\textsc{DProQ} achieves the best hit rate of $12\/10\/3$ compared to all other methods. Additionally, \\textsc{DProQ} successfully selects all $10$ targets with Medium quality decoys and all $3$ targets with High quality decoys, moreover hitting all three accuracy levels' decoys on $6$ targets which is the best result achieved compared to other methods. On this dataset, GNN\\_DOVE achieves a $10\/4\/1$ hit rate, worse than all other methods. Nonetheless, GNN\\_DOVE successfully hits two targets, \\textsc{5WK3} and \\textsc{5KOV}.\n\nTable \\ref{tab:dbm_ranking} presents the ranking loss for all methods on DBM55-AF2 dataset. Here, \\textsc{DProQ} achieves the best ranking loss of $0.049$ which is $87\\%$ lower than GNN\\_DOVE's ranking loss of $0.379$. Furthermore, for $4$ targets, \\textsc{DProQ} correctly selects the Top-$1$ model and achieves $0$ ranking loss. Similarly to their results on the HAF2 dataset, \\textsc{DProQ\\_GT}, \\textsc{DProQ\\_GTE}, and \\textsc{DProQ\\_GTN}'s losses are $71\\%$, $84\\%$, and $74\\%$ lower, respectively, than GNN\\_DOVE's ranking loss.\n\n\\begin{table}\n\\label{tab:dbm_hits}\n\\centering\n\\caption{Hit rate performance on DBM55-AF2 dataset. The \\textsc{BEST} column represents each target's best-possible Top-$10$ result. The \\textsc{Summary} row lists the results when all targets are taken into consideration.}\n\\begin{tabular}{lllllll}\n\\toprule\nTarget & \\textsc{DProQ} & \\textsc{DProQ}\\_GT & \\textsc{DProQ}\\_GTE & \\textsc{DProQ}\\_GTN & GNN\\_DOVE & \\textsc{BEST} \\\\ \n\\midrule\n6AL0 & 9\/2\/0 & 10\/0\/0 & 10\/0\/0 & 10\/2\/0 & 6\/0\/0 & 10\/2\/0 \\\\\n3SE8 & 8\/8\/0 & 9\/9\/0 & 8\/8\/0 & 8\/8\/0 & 3\/0\/0 & 10\/10\/0 \\\\\n5GRJ & 10\/10\/0 & 9\/9\/0 & 10\/10\/0 & 9\/9\/0 & 3\/2\/0 & 10\/10\/0 \\\\\n6A77 & 7\/7\/0 & 7\/7\/0 & 8\/8\/0 & 8\/8\/0 & 0\/0\/0 & 8\/8\/0 \\\\\n4M5Z & 10\/10\/1 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/1 \\\\\n4ETQ & 1\/1\/0 & 1\/1\/0 & 1\/1\/0 & 1\/1\/0 & 0\/0\/0 & 1\/1\/0 \\\\\n5CBA & 10\/10\/1 & 10\/10\/0 & 10\/10\/0 & 10\/10\/1 & 10\/10\/3 & 10\/10\/6 \\\\\n5WK3 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 1\/0\/0 & 3\/0\/0 \\\\\n5Y9J & 4\/0\/0 & 6\/0\/0 & 5\/0\/0 & 4\/0\/0 & 0\/0\/0 & 8\/0\/0 \\\\\n6BOS & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 & 10\/10\/0 \\\\\n5HGG & 8\/0\/0 & 8\/0\/0 & 8\/0\/0 & 8\/0\/0 & 8\/0\/0 & 10\/0\/0 \\\\\n6A0Z & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 2\/0\/0 & 3\/0\/0 \\\\\n3U7Y & 2\/2\/1 & 2\/2\/1 & 2\/2\/1 & 2\/1\/0 & 2\/2\/1 & 2\/2\/1 \\\\\n3WD5 & 10\/8\/0 & 9\/8\/0 & 9\/8\/0 & 9\/8\/0 & 0\/0\/0 & 10\/10\/0 \\\\\n5KOV & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 0\/0\/0 & 1\/0\/0 & 2\/0\/0 \\\\\n\\midrule\n\\textsc{Summary} & \\textbf{12\/10\/3} & 12\/9\/1 & 12\/9\/1 & 12\/10\/1 & 10\/4\/1 & 15\/10\/3 \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\label{tab:dbm_ranking}\n\\centering\n\\caption{Ranking loss performance on the DBM55-AF2 dataset. The \\textsc{BEST} row represents the mean and standard deviation of the ranking losses for all targets.}\n\\begin{tabular}{llllll}\n\\toprule\nTarget & \\textsc{DProQ} & \\textsc{DProQ}\\_GT & \\textsc{DProQ}\\_GTE & \\textsc{DProQ}\\_GTN & GNN\\_DOVE \\\\\n\\midrule\n6AL0 & 0.0 & 0.156 & 0.156 & 0.0 & 0.424 \\\\\n3SE8 & 0.079 & 0.041 & 0.041 & 0.079 & 0.735 \\\\\n5GRJ & 0.024 & 0.012 & 0.095 & 0.012 & 0.776 \\\\\n6A77 & 0.037 & 0.062 & 0.0 & 0.037 & 0.591 \\\\\n4M5Z & 0.015 & 0.026 & 0.026 & 0.015 & 0.221 \\\\\n4ETQ & 0.0 & 0.76 & 0.0 & 0.748 & 0.759 \\\\\n5CBA & 0.052 & 0.038 & 0.052 & 0.058 & 0.019 \\\\\n5WK3 & 0.114 & 0.114 & 0.114 & 0.186 & 0.087 \\\\\n5Y9J & 0.0 & 0.0 & 0.0 & 0.0 & 0.382 \\\\\n6BOS & 0.081 & 0.081 & 0.0 & 0.0 & 0.081 \\\\\n5HGG & 0.051 & 0.051 & 0.121 & 0.051 & 0.121 \\\\\n6A0Z & 0.207 & 0.207 & 0.207 & 0.207 & 0.062 \\\\\n3U7Y & 0.0 & 0.021 & 0.0 & 0.0 & 0.756 \\\\\n3WD5 & 0.011 & 0.011 & 0.011 & 0.0 & 0.672 \\\\\n5KOV & 0.065 & 0.08 & 0.085 & 0.087 & 0.0 \\\\\n\\midrule\n\\textsc{BEST} & \\textbf{0.049 \u00b1 0.054} & 0.111 \u00b1 0.182 & 0.061 \u00b1 0.064 & 0.099 \u00b1 0.185 & 0.379 \u00b1 0.298 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\textbf{Node and Edge Gates.} In Tables \\ref{tab:haf2_hits}, \\ref{tab:haf2_ranking}, \\ref{tab:dbm_hits}, and \\ref{tab:dbm_ranking}, we evaluated \\textsc{DProQ\\_GT}, \\textsc{DProQ\\_GTE}, and \\textsc{DProQ\\_GTN}'s performance on our two test datasets to discover the node and edge gates' effects on performance. On the HAF2 dataset, our results indicate that exclusively adding either a node or edge gate can help improve a model's Top-$1$ structure ranking ability.\n\nOur gate-ablation results yielded another interesting phenomenon when we observed that \\textsc{DProQ\\_GTE} consistently achieved the best ranking loss on both of our test datasets, compared to \\textsc{DProQ\\_GTN}. Notably, on the DBM55-AF2 dataset, \\textsc{DProQ\\_GTE}'s loss is much lower than that of other methods, except for \\textsc{DProQ}. This suggests \\textsc{DProQ\\_GTE} is more sensitive to Top-$1$ structure selection compared to \\textsc{DProQ\\_GTN}. We hypothesize this trend may be because, within \\textsc{DProQ}, k-NN graphs' edges may not always possess a biological interpretation during graph message passing. In such cases, models' edge gates may allow the network to inhibit the weight of biologically-implausible edges, which could remove noise from the neural network's training procedure. We plan to investigate extensions of the GGT to explore these ideas further.\n\n\\textbf{Repeated Experiments.} In addition, we trained four more \\textsc{DProQ} models within the original \\textsc{DProQ} training environment but with different random seeds. We see in Appendix \\ref{sec:appendix_a} that \\textsc{DProQ} demonstrates consistently-better results compared to GNN\\_DOVE. We observe \\textsc{DProQ} providing reliable, state-of-the-art performance in terms of both hit rate and ranking loss on both test datasets.\\par\n\n\n\n\n\\textbf{Limitations.} One limitation of the \\textsc{GGT} is that its architecture does not directly operate using protein spatial information. Models such as the SE(3)-Transformer \\cite{fuchs2020se} may be one way to explore this spatial information gap. However, due to such models' high computational costs, we are currently unable to utilize such techniques within the context of protein complex structure modeling. Moreover, in our studies, \\textsc{DProQ} failed on some test targets that contain few Acceptable or higher-quality decoys within the Top-$10$ range. To overcome this problem, we experimented with multi-tasking within \\textsc{DProQ}, but we did not find such multi-tasking to help our models generalize better. We hypothesize that new, diverse datasets may be helpful in this regard. We defer such ideas to future work.\n\n\\subsection{Gated Graph Transformer architecture}\n\\label{sec:gate_gt}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{images\/gated_graph_transformer.png}\n \\caption{\\textsc{GGT} model architecture.}\n \\label{fig:gate_gt}\n\\end{figure}\n\nWe designed the Gated Graph Transformer (\\textsc{GGT}) as a solution to a specific phenomenon in graph representation learning. Note that, unlike other GNN-based structure scoring methods, \\cite{wang2021protein, hanquality} define edges using a fixed distance threshold, which means that each graph node may have a different number of incoming and outgoing edges. In contrast, \\textsc{DProQ} constructs and operates on k-NN graphs where all nodes are connected to the same number of neighbors. However, in the context of k-NN graphs, each neighbor's information is, by default, given equal priority during information updates. As such, we may desire to imbue our graph neural network with the ability to automatically decide the priority of different nodes and edges during the graph message passing. Consequently, we present the \\textsc{GGT}, a gated neighborhood-modulating Graph Transformer inspired by \\cite{velivckovic2017graph, dwivedi2020generalization, morehead2021geometric}. Formally, to update the network's node embeddings $\\mathbf{h}_{i}$ and edge embeddings $\\mathbf{e}_{ij}$, we define a single layer of the \\textsc{GGT} as:\n\n\\begin{equation}\n\\label{eq:w2}\n \\widehat{\\widehat{\\mathbf{w}}}_{ij}^{k, \\ell}=\\left(\\frac{Q^{k, \\ell} \\mathbf{h}_{i}^{\\ell} \\cdot K^{k, \\ell} \\mathbf{h}_{j}^{\\ell}}{\\sqrt{d_{k}}}\\right) \\cdot E^{k, \\ell} \\mathbf{e}_{i j}^{\\ell}\n\\end{equation}\n\n\\begin{equation}\n \\label{eq:e_update}\n \\hat{\\mathbf{e}}_{ij}^{\\ell+1}=\\mathbf{O}_{e}^{\\ell} \\|_{k=1}^{H}\\left(\\widehat{\\widehat{\\mathbf{w}}}_{ij}^{k, \\ell}\\right)\n\\end{equation}\n\n\\begin{equation}\n \\label{eq:w1}\n \\widehat{\\mathbf{w}}_{ij}^{k, \\ell}=\\widehat{\\widehat{\\mathbf{w}}}_{ij}^{k, \\ell} \\times \\operatorname{sigmoid}\\left(\\mathbf{G}_{e}^{k, \\ell} \\mathbf{e}_{i j}^{\\ell}\\right)\n\\end{equation}\n\n\\begin{equation}\n \\label{eq:w}\n \\mathbf{w}_{ij}^{k, \\ell}=\\operatorname{softmax}\\left(\\hat{\\mathbf{w}}_{ij}^{k, \\ell}\\right)\n\\end{equation}\n\n\\begin{equation}\n \\label{eq:h_update}\n \\hat{\\mathbf{h}}_{i}^{\\ell+1}=O_{\\mathbf{h}}^{\\ell} \\|_{k=1}^{H}\\left(\\operatorname{sigmoid}\\left(G_{h}^{k, \\ell} \\mathbf{h}_{j}^{\\ell}\\right) \\times \\sum_{j \\in \\mathcal{N}_{i}} \\mathbf{w}_{i j}^{k, \\ell} V^{k, \\ell} \\mathbf{h}_{j}^{\\ell}\\right)\n\\end{equation}\n\nIn particular, the \\textsc{GGT} introduces to the standard Graph Transformer architecture \\cite{dwivedi2020generalization} two information gates through which the network can modulate node and edge information flow, as shown in Figure \\ref{fig:gate_gt}. Equation \\ref{eq:e_update} shows how the output of the edge gate, $\\mathbf{G}_{e}$, is used elementwise to gate the network's attention scores $\\widehat{\\widehat{\\mathbf{w}}}_{ij}$ derived from Equation \\ref{eq:w2}. Such gating simultaneously allows the network to decide how much to regulate edge information flow as well as how much to weigh the edge of each node neighbor. Similarly, the node gate, $\\mathbf{G}_{h}$, shown in Equation \\ref{eq:h_update}, allows the GGT to individually decide how to modulate information from node neighbors during node information updates. Lastly, for the GGT's Feed Forward Network, we keep the same structure as described in \\cite{dwivedi2020generalization}, providing us with new node embeddings $\\hat{\\mathbf{h}}_{i}^{L}$ and edge embeddings $\\hat{\\mathbf{e}}_{ij}^{L}$. For additional background on the GGT's network operations, we refer readers to \\cite{dwivedi2020generalization}.\n\n\\subsection{Multi-task graph property prediction}\n\\label{sec:graph_predictions}\nTo obtain graph-level predictions for each input protein complex graph, we apply a graph sum-pooling operator on $\\hat{\\mathbf{h}}_{i}^{L}$ such that $\\mathbf{p} \\in \\mathbb{R}^{d \\times 1} = \\sum_{i = 1}^{N} \\mathbf{h}_{i}^{L}$. This graph embedding $\\mathbf{p}$ is then fed as input to \\textsc{DProQ}'s first read-out module $\\varphi_{1}^{r}$, where each read-out module consists of a series of linear layers, batch normalization layers, LeakyReLU activations, and dropout layers \\cite{hinton2012improving}, respectively. That is, $\\mathbf{\\hat{p}} \\in \\mathbb{R}^{d \\times 4} = \\varphi_{1}^{r}(\\mathbf{p})$ reduces the dimensionality of $\\mathbf{p}$ to accommodate \\textsc{DProQ}'s output head designed for graph classification. Specifically, for graph classification, we apply a Softmax layer such that $\\mathbf{y} \\in \\mathbb{R}^{d \\times 4} = softmax(\\mathbf{\\hat{p}})$, where $\\mathbf{y}$ is the network's predicted DockQ structure quality class probabilities \\cite{basu2016dockq} for each input protein complex (e.g., Medium, High). Thereafter, we apply \\textsc{DProQ}'s second read-out module $\\varphi_{2}^{r}$ to $\\mathbf{\\hat{p}}$ such that $\\mathbf{q} \\in \\mathbb{R}^{1 \\times 1} = \\varphi_{2}^{r}(\\mathbf{\\hat{p}})$. Within this context, the network's scalar graph regression output $\\mathbf{q}$ represents the network's predicted DockQ score \\cite{basu2016dockq} for a given protein complex input.\n\n\\textbf{Structure quality scoring loss.} To train \\textsc{DProQ}'s graph regression head, we used the mean squared error loss $\\mathcal{L}_{R} = \\frac{1}{N} \\sum_{i = 1}^{N} \\| \\mathbf{q}_{i}' - \\mathbf{q}_{i}^{*} \\|^{2}$. Here, $\\mathbf{q}_{i}'$ is the model's predicted DockQ score for example $i$, $\\mathbf{q}_{i}^{*}$ is the ground truth DockQ score for example $i$, and $N$ represents the number of examples in a given mini-batch.\n\n\\textbf{Structure quality classification loss.} Likewise, to train \\textsc{DProQ}'s graph classification head, we used the cross-entropy loss $\\mathcal{L}_{C} = \\frac{1}{N} \\sum_{i = 1}^{N} \\left(\\mathbf{y}_{i}^{*} - \\log(\\mathbf{y}_{i}')\\right)$. Here, $\\mathbf{y}_{i}'$ is the model's predicted DockQ quality class (e.g., Acceptable) for example $i$, and $\\mathbf{y}_{i}^{*}$ is the ground truth DockQ quality class (e.g., Incorrect) for example $i$.\n\n\\textbf{Overall loss.} We define \\textsc{DProQ}'s overall loss as $\\mathcal{L} = w_{\\mathcal{L}_{C}} \\times \\mathcal{L}_{C} + w_{\\mathcal{L}_{R}} \\times \\mathcal{L}_{R}$. We note that the weights for each constituent loss (e.g., $w_{\\mathcal{L}_{R}}$) were determined either by performing a grid search or instead by using a lowest validation loss criterion for parameter selection. In Appendix \\ref{sec:appendix_c}, we describe \\textsc{DProQ}'s choice of hyperparameters in greater detail.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:Introduction}Introduction}\n\nRecently, a new state of matter, the topological insulator,\nattracts much theoretical and experimental\nattention.\\cite{kane05,bernevig06,sheng06,konig07,moore07,roy06,fu07l,fu07b,\nhsieh08,qi08,xia09,zhang09,hasan10rmp,qi10rev,schnyder08} It\nhas an electronic structure dominated by the spin-orbit\ncoupling, which is a band insulator with a well-defined gap in\nthe bulk but can host an odd number of Dirac cones protected by\ntime reversal symmetry on the\nsurface.\\cite{kane05,bernevig06,fu07l} The intrinsic spin-orbit\ncoupling makes it promising for spintronics\napplications.\\cite{qi08} The proximity induced superconducting\nstate on the topological insulator surface by a deposited\nsuperconductor was proposed to create Majorana\nfermions\\cite{wilczek09}, which may provide a new way to\nrealize the topological quantum computation.\\cite{fu0809,sau10}\nLately, the realization of a superconducting state in a typical\ntopological insulator Bi$_{2}$Se$_{3}$ by intercalating Cu\nbetween adjacent quintuple units (Cu$_{x}$Bi$_{2}$Se$_{3}$)\nmakes the system even more attractive.\\cite{hor10,wray10} The\nlarge diamagnetic response shows that the pairing is mainly of\nbulk character. In another topological insulator\nBi$_{2}$Te$_{3}$, the application of a high pressure also turns\nthe material into a superconducting\nstate.\\cite{zhang10pa,zhang10pb}\n\nSince the superconductivity is bulk and intrinsic to the\nmaterial, if zero energy surface Majorana fermion mode exists,\nit would be easier to manipulate as compared to that induced by\nproximity effect, as it would not experience the interface\nroughness or mismatch common to a junction type device.\nPossible nontrivial odd-parity pairing in\nCu$_{x}$Bi$_{2}$Se$_{3}$ is proposed and analyzed by Fu and\nBerg.\\cite{fu10} It was argued that only if a bulk gap opens\nand the bulk pairing is odd in parity, would zero energy\nAndreev bound states appear in the surface spectrum. However,\ntheir analysis was concentrated on the case when the chemical\npotential is much larger than the gap and the topological\nsurface states are already merged into the continuum conduction\nband. A similar analysis was put forward by\nSato.\\cite{sato0910}\n\nDespite the above works, a detailed theoretical analysis of\nsurface spectrum in the superconducting phase arising from a\ntopological insulator is still lacking. In particular, not much\nis reported for the situation when the chemical potential is\nonly slightly larger than the insulating gap and both\ntopological surface states (or, the surface conduction\nband\\cite{wray10}) and the continuum bulk conduction band are\npresent but separated. Since the surface spectrum is central to\nthe topological properties of a material both in the normal and\nin the superconducting state, which is also directly accessible\nby experimental techniques as ARPES\\cite{wray10} and\nSTM\\cite{alpichshev10}, it is highly desirable to make a\ndetailed study of them. This will help to understand better\nsuperconductivity in systems with nontrivial topological band\nstructure.\n\nWe focus on two questions in this paper. One is the effect of\nsuperconducting pairing on the topological surface\nstates\\cite{wray10} present in the normal state. The other is\nthe existence of surface Andreev bound states. We have noticed\nthat two different models\\cite{zhang09,wang10} are often used\nin-discriminatively in literature. They have the same normal\nstate energy spectra but may be different in the\nsuperconducting state. We thus present our results for both of\nthem. What happens to the topological surface states when\npairing is introduced in the bulk depends on the orbital\ncharacter of the topological surface state and on the bulk\npairing symmetry. Only when the continuum part of the band\nopens a full gap and the topological surface states, while\nseparated from the bulk conduction band, do not open a gap for\nan odd-parity pairing, would a gapless Andreev bound state\nappear. We find that the orbital characters of the topological\nsurface states are different for the two models. For a certain\nbulk pairing symmetry, it is possible that the topological\nsurface states of one model opens a gap while that of the other\nmodel is still intact. The existence of Andreev bound states,\nthe most important indication of nontrivial topological order\nin the superconducting phase, is thus expected to be also\nrelated to the orbital character of the topological surface\nstates. We show that the interplay between the continuum bulk\nconduction band and the topological surface states produces a\nring or a segment of zero energy states in addition to the\nAndreev bound states depending on the symmetry of bulk pairing\norder parameters.\n\n\n\n\n\\section{models and the normal state surface modes}\n\nIn the following, when we talk about topological insulators, we\nwould be mainly referring to Bi$_2$Se$_3$, which shows a well\ndefined Dirac cone structure for the topological surface\nstates.\\cite{zhang09,xia09,hsieh09} The model we consider below\ncould be easily generalized to study other topological\ninsulators like TlBiSe$_2$\\cite{yan10,kuroda10} and\nBi$_{2}$Te$_{2}$Se\\cite{ren10,xu10}.\n\nThe band structure of Cu$_{x}$Bi$_{2}$Se$_{3}$ is similar to\nthat of Bi$_{2}$Se$_{3}$, the most essential part of which\nconsists of two $p_z$ orbitals on the top and bottom Se layers\nhybridized with neighboring Bi $p_z$ orbitals, in each\nquintuple Bi$_{2}$Se$_{3}$ unit.\\cite{zhang09,fu10} In the\npresence of spin-orbit coupling, the normal state has four\ndegrees of freedom. Label the two orbitals concentrating mainly\non the top and bottom (seeing along the $-z$ direction) Se\nlayer of the various Bi$_2$Se$_3$ quintuple units as the first\nand second orbital, the basis is taken as\n$\\psi_{\\mathbf{k}}$=[$c_{1\\mathbf{k}\\uparrow}$,\n$c_{2\\mathbf{k}\\uparrow}$, $c_{1\\mathbf{k}\\downarrow}$,\n$c_{2\\mathbf{k}\\downarrow}$]$^{T}$. The models could be written\ncompactly in the following matrix\nform\\cite{zhang09,wang10,fu10}\n\\begin{equation} \\label{h3d0}\nH(\\mathbf{k})=\\epsilon_{0}(\\mathbf{k})I_{4\\times4}+\\sum\\limits_{i=0}^{3}m_{i}(\\mathbf{k})\\Gamma_{i}.\n\\end{equation}\nI$_{4\\times4}$ is the fourth order unit matrix, giving rise to\na topologically trivial shift of the energy bands and would be\nneglected in most of our following analysis. In terms of the\ntwo by two Pauli matrices $s_{i}$ ($i$=0, $\\cdots$, 3) in the\nspin subspace and $\\sigma_{i}$ ($i$=0, $\\cdots$, 3) in the\norbital subspace, the first three Dirac matrices are defined\nas\\cite{zhang09,wang10} $\\Gamma_{0}$=$s_{0}\\otimes\\sigma_{1}$,\n$\\Gamma_{1}$=$s_{1}\\otimes\\sigma_{3}$,\n$\\Gamma_{2}$=$s_{2}\\otimes\\sigma_{3}$. As regards $\\Gamma_{3}$,\nthere are presently two different choices:\n(I)$s_{0}\\otimes\\sigma_{2}$\\cite{wang10,fu10,li10} and\n(II)$s_{3}\\otimes\\sigma_{3}$\\cite{zhang09,liu10b1,liu10b2,lu10,shan10},\nwhich define the two models that would be considered in\nparallel in the following discussion. Since Bi$_{2}$Se$_{3}$ is\ninversion symmetric, parity could be used to label states. Some\npapers adopt the bonding and antibonding states of the two\norbitals defined above as the orbital\nbasis.\\cite{zhang09,li10,liu10b1,liu10b2,lu10,shan10} The\ncorresponding models could be obtained in terms of a simple\nunitary transformation performed in the orbital subspace. Note\nthat, the two different models give the same bulk band\ndispersion\n$\\epsilon_{\\pm}(\\mathbf{k})$=$\\epsilon_{0}(\\mathbf{k})\\pm\\sqrt{\\sum_{i=0}^{3}m_{i}^{2}(\\mathbf{k})}$,\nwith each of two eigen-energies two fold Kramers degenerate due\nto the time reversal symmetry and the inversion symmetry of the\nmodel. However, we will show in the following that they are\nessentially two physically distinct models.\n\n\n\nFor the coefficients $\\epsilon_{0}(\\mathbf{k})$ and\n$m_{i}(\\mathbf{k})$ ($i=$ 0, 1, 2, 3), there are different\npossible parameterizations which coincide with each other close\nto the $\\Gamma$ point.\\cite{zhang09,qi10rev,fu09,wang10}\nWithout loss of generality, we take the parameterizations of\nWang \\emph{et al.}\\cite{wang10}. Since the diagonal term\n$\\epsilon_{0}(\\mathbf{k})$ proportional to the unit matrix does\nnot affect the topological characters, it is ignored here. The\nsystem is defined on a hypothetical bilayer hexagonal lattice\nstacked along the $z$-axis, respecting the in plane hexagonal\nsymmetry of the original Bi$_{2}$Se$_{3}$ lattice. With the\nthree independent in-plane nearest neighbor unit vectors\ndefined as $\\hat{b}_1$=($\\frac{\\sqrt{3}}{2}$, $\\frac{1}{2}$),\n$\\hat{b}_2$=(-$\\frac{\\sqrt{3}}{2}$, $\\frac{1}{2}$), and\n$\\hat{b}_3$=(0, -1), we have\n$m_{0}(\\mathbf{k})$=$m+2t_{z}(1-\\cos\nk_{z})+2t(3-2\\cos\\frac{\\sqrt{3}}{2}k_{x}\\cos\\frac{1}{2}k_{y}-\\cos\nk_{y})$,\n$m_{1}(\\mathbf{k})$=2$\\sqrt{3}t\\sin\\frac{\\sqrt{3}}{2}k_{x}\\cos\\frac{1}{2}k_{y}$,\n$m_{2}(\\mathbf{k})$=2$t(\\cos\\frac{\\sqrt{3}}{2}k_{x}\\sin\\frac{1}{2}k_{y}+\\sin\nk_{y})$, and $m_{3}(\\mathbf{k})$=$2t_{z}\\sin k_{z}$. The\nin-plane and out-of-plane lattice parameters\\cite{larson02} are\ntaken as length units in the above expression, that is\n$a$=$c$=1. When $mt_z$$<$0 and $mt$$<$0, the parametrization\ndefined above\\cite{wang10} and the parametrization in the small\n$\\mathbf{k}$ effective model proposed by Zhang \\emph{et\nal.}\\cite{zhang09} describe qualitatively the same physics.\nWith this parametrization, it is easy to see that the model has\nthe inversion symmetry $PH(\\mathbf{k})P^{-1}$=$H(-\\mathbf{k})$,\nwhere the inversion operator is defined as\n$P$=$s_0\\otimes\\sigma_1$.\\cite{fu07b}\n\nNow, we clarify the differences between models (I) and (II) by\ntheir surface states, which is one of the most important\nsignatures of nontrivial topological order in the system. Close\nto the $\\Gamma$ point in the BZ, we take $m_{i=0, \\cdots,\n3}(\\mathbf{k})$=$\\{m+\\frac{3}{2}t(k_{x}^2+k_{y}^2)+t_{z}k_{z}^{2},\n3tk_{x}, 3tk_{y}, 2t_{z}k_{z}\\}$, in which $t>0$, $t_{z}>0$ and\n$m<0$. Consider a sample occupying the lower half space\n$z\\le0$. The possible surface states localized close to $z=0$\nis searched by solving a set of four coupled second order\ndifferential equations\n\\begin{equation}\nH(k_{x}=k_{y}=0,k_{z}\\rightarrow -i\\partial_{z})\\Psi(z)=E\\Psi(z),\n\\end{equation}\ntogether with the open boundary condition\n$\\Psi(z)|_{z=0}$=$\\Psi(z)|_{z=-\\infty}$=0.\\cite{liu10b2,lu10}\n$\\Psi(z)$ is the four-component eigenvector and E is the energy\nof the surface mode, respectively. We look for the zero energy\nstates and hence set $E$=0.\\cite{liu10b2}\n\nFor the model (I) with $\\Gamma_3$=$s_{0}$$\\otimes$$\\sigma_{2}$,\nthe up and down spin degrees of freedom are decoupled from each\nother. The wave function could thus be written as\n$\\Psi(z)$=$[u_{1}(z), u_{2}(z), u_{3}(z),\nu_{4}(z)]^{T}$=$[\\chi_{\\uparrow}(z),\n\\chi_{\\downarrow}(z)]^{T}$. The two spin components of the zero\nenergy mode satisfy the same equation as ($s$ is $\\uparrow$ or\n$\\downarrow$ for the two spin degrees of freedom)\n\\begin{equation}\n[(m-t_{z}\\partial_{z}^{2})\\sigma_{1}-2it_{z}\\partial_{z}\\sigma_{2}] \\chi_{s}(z)=0.\n\\end{equation}\nThe two degenerate zero energy surface modes for $z$$\\le$0 are\nobtained as\n\\begin{equation}\n\\Psi_{\\alpha}(z)=C\\eta_{\\alpha}(e^{z\/\\xi_{+}}-e^{z\/\\xi_{-}}),\n\\end{equation}\nwhere $\\alpha$=1 or 2, $C$ is a normalization constant and\n$\\xi_{\\pm}^{-1}$=$1\\pm\\sqrt{1+m\/t_{z}}$. 1\/Re$[\\xi_{\\pm}^{-1}]$\n(`Re' means taking the real part of a number) are the two\npenetration depths of the surface modes into the bulk. The two\nunit vectors are $(\\eta_{1})_{\\beta}$=$\\delta_{\\beta1}$ and\n$(\\eta_{2})_{\\beta}$=$\\delta_{\\beta3}$, where\n$\\delta_{\\alpha\\beta}$ is one for $\\alpha$=$\\beta$ and zero\notherwise. Take $\\{\\Psi_{1}, \\Psi_{2}\\}$ as the two basis, the\neffective model for the surface states are obtained by\nconsidering the k$_{x}$ and k$_{y}$ dependent terms in the\noriginal model as perturbations, which are\n\\begin{equation}\n\\Delta H_{3D}=\\frac{3}{2}t(k_{x}^2+k_{y}^2)\\Gamma_{0}+3t(k_{x}\\Gamma_{1}+k_{y}\\Gamma_{2}).\n\\end{equation}\nSuppose the two basis are normalized, the effective model for\nthe surface states is\\cite{zhang09}\n\\begin{equation}\nH_{eff}(\\mathbf{k})=3t(k_{x}s_{x}+k_{y}s_{y}),\n\\end{equation}\nwhere $s_{x}$ and $s_{y}$ are the first and second Pauli\nmatrices. Since the two basis both have definite spin\ncharacters, $s_{x}$ and $s_{y}$ in the above equation could\nalso be considered as acting in the spin subspace. The most\nsalient feature of this model is that the corresponding surface\nstates has contributions only from the first orbital. When we\nconsider a sample occupying $z$$\\ge$0, the surface states at\n$z$=0 would arise only from the second orbital.\n\n\nWe now study model (II) for\n$\\Gamma_{3}$=$s_{3}$$\\otimes$$\\sigma_{3}$. We still consider a\nsample situated at $z$$\\le$0 with the open boundary conditions.\nFollowing exactly the same steps as for the first model, we\nobtain the two degenerate zero energy surface states as\n\\begin{equation}\n\\Phi_{\\alpha}=C\\eta_{\\alpha}(e^{z\/\\xi_{+}}-e^{z\/\\xi_{-}}),\n\\end{equation}\nwhere $\\alpha$=1 or 2, $C$ is a normalization constant and\n$\\xi_{\\pm}$ are defined identically as above. However, the two\nunit basis vectors are quite different from the first model and\nare $\\eta_{1}$=$\\frac{1}{\\sqrt{2}}[1, -i, 0, 0]^{T}$ and\n$\\eta_{2}$=$\\frac{1}{\\sqrt{2}}[0, 0, -i, 1]^{T}$. The two\ndimensional effective model for the surface states is also a\nbit different at least formally, which is\n\\begin{equation}\nH_{eff}(\\mathbf{k})=3t\\hat{z}\\cdot(\\mathbf{k}\\times\\mathbf{s})=3t(k_{x}s_{y}-k_{y}s_{x}).\n\\end{equation}\nThe Pauli matrices $s_{x}$ and $s_{y}$ act in the two fold\ndegenerate basis of the zero energy surface states which both\nhave definite spin characters. For a general two dimensional\nwave vector, the surface state would be a linear combination of\nall the four spin-orbital basis.\n\nThus there are qualitative differences between two models which\nwere used in-discriminatively in the literature for\nBi$_{2}$Se$_{3}$.\\cite{wang10,fu10,li10,zhang09,liu10b1,liu10b2,lu10,shan10}\nWhile only one orbital contributes to the surface states for\nmodel (I), both two orbitals contribute in equal weight to the\nsurface states for model (II). On the other hand, the effective\nmodel of the surface states has the same spin-orbital coupled\nform as the linear $k_{x}$ and $k_{y}$ terms in the original\nthree dimensional model for model (I). However, the effective\nmodel is changed from $\\mathbf{k}\\cdot\\mathbf{s}$ to\n$\\hat{z}\\cdot(\\mathbf{k}\\times\\mathbf{s})$ for model (II). We\nhave verified that, if we change the in-plane spin-orbit\ncoupling of the two models (I) and (II) from\n$\\mathbf{k}\\cdot\\mathbf{s}$ to\n$\\hat{z}\\cdot(\\mathbf{k}\\times\\mathbf{s})$, the resulting\neffective model of the surface states would have the form\n$\\hat{z}\\cdot(\\mathbf{k}\\times\\mathbf{s})$ for model (I) but\nwill be switched to $\\mathbf{k}\\cdot\\mathbf{s}$ for model (II).\n\n\nBefore ending this section, we would like to point out that\nboth the $k_{z}$-linear term in $m_{3}(\\mathbf{k})$ and the\n$k_{z}$-square term in $m_{0}(\\mathbf{k})$ are essential to\nobtain the zero energy surface modes. If we omit the\n$k_{z}^{2}$ term in $m_{0}(\\mathbf{k})$, it is easy to verify\nthat the gapless surface states no longer exist. This is\nrelated to the fact that band inversion is essential to the\nappearance of nontrivial topological surface\nstates\\cite{bernevig06,qi10pto}, which can only occur in the\npresence of $k_{z}^{2}$ term for $k_{x}$=$k_{y}$=0.\n\n\n\n\\section{superconducting state spectral function}\n\n\\subsection{surface Green's functions in the superconducting state}\n\nThe realization of the superconducting state in\nCu$_{x}$Bi$_{2}$Se$_{3}$\\cite{hor10,wray10} has brought about\nexcitement that non-trivial topological superconducting state\nmight be realized in this system, in which topologically\nprotected gapless surface states traverse the bulk\nsuperconducting gap.\\cite{fu10,sato0910,qi10b} The recent\nrealization of superconducting phase in Bi$_{2}$Te$_{3}$ under\nhigh pressure\\cite{zhang10pa,zhang10pb} makes the\nBi$_{2}$X$_{3}$ (X is Se or Te) material a very promising\ncandidate system to realize topologically nontrivial\nsuperconducting phases.\\cite{sato0910,fu10,qi10b}\n\n\nThe normal state of the topological insulator is marked by the\npresence of topological surface states inside the bulk gap.\nThese gapless topological surface states are well separated\nfrom the bulk conduction band at low energies and become\nindistinguishable for energies much higher than the conduction\nband minimum. Depending on the doped charge density the\nsuperconducting state could occur with the chemical potential\neither deep in the bulk conduction band or in the intermediate\nregion where the topological surface states are well separated\nfrom the bulk conduction band\\cite{wray10}. In the latter case,\nthe coupling between the continuum bulk states and the isolated\ntopological surface states may cause some new interesting\nphenomena. Thus this intermediate region is where we will\nconcentrate on below. Furthermore, the actual pairing\nsymmetries of the superconducting Cu$_{x}$Bi$_{2}$Se$_{3}$ and\nBi$_{2}$Te$_{3}$ are presently\nunknown.\\cite{hor10,wray10,zhang10pa,zhang10pb} Thus we will\nexamine cases with different pairing symmetries in the hope to\nprovide clues to identify the pairing symmetry and the role\ncontributed by the topological surface states.\n\nIn the following, we will study the surface spectral function\nto see possible nontrivial topological properties arising from\nthe normal phase topological order which is subject to a\ncertain bulk pairing. The surface spectral function, which\ncould be obtained from the surface Green's function, has been\nstudied by ARPES\\cite{wray10} and STM\\cite{alpichshev10} to\ngive important information on the topological properties of the\nsystem. In the superconducting state, we expect to see some\nsurface Andreev bound states if a certain superconducting order\nis realized in the material.\n\nIn the presence of a surface perpendicular to the $z$-axis,\n$k_{x}$ and $k_{y}$ are good quantum numbers, and $k_{z}$ is\nreplaced by $-i\\partial_{z}$ as we shall search for surface\nstates. We then discretize the $z$ coordinate and turn the\nwhole sample ($z$$\\le$0) with a surface at $z$=0 to a coupled\nquintuple-layer system. Label each separate quintuple unit with\nan integer index $n$, and make the substitutions\n$\\partial_{z}\\psi_{n}(z)$=$\\frac{1}{2}[\\psi_{n+1}-\\psi_{n-1}]$\nand\n$\\partial_{z}^{2}\\psi_{n}(z)$=$\\psi_{n+1}+\\psi_{n-1}-2\\psi_{n}$\n($c$ is set as length unit along $z$ axis), the Hamiltonian\nconsists now the intra-layer terms and the interlayer hopping\nterms, $\\hat{H}$=$\\hat{H}_{\\parallel}+\\hat{H}_{\\perp}$. The\nintra-layer part of the model is\n$\\hat{H}_{\\parallel}$=$\\sum_{n\\mathbf{k}}\n\\psi_{n\\mathbf{k}}^{\\dagger}h_{xy}(\\mathbf{k})\\psi_{n\\mathbf{k}}$,\nin which\n\\begin{equation}\nh_{xy}(\\mathbf{k})=m'_{0}(\\mathbf{k})\\Gamma_{0}+m_{1}(\\mathbf{k})\\Gamma_{1}\n+m_{2}(\\mathbf{k})\\Gamma_{2}.\n\\end{equation}\n$m_{1}(\\mathbf{k})$ and $m_{2}(\\mathbf{k})$ are the same as\nthose in the bulk model. $m'_{0}(\\mathbf{k})$ is obtained from\n$m_{0}(\\mathbf{k})$ by first expanding it up to the square term\nof $k_{z}$ and then replacing the term proportional to\n$k_{z}^{2}$ from $Ck_{z}^{2}$ to $2C$.\n\nThe inter-layer hopping term is\n$\\hat{H}_{\\perp}$=$\\sum_{n\\mathbf{k}}\\psi_{n\\mathbf{k}}^{\\dagger}h_{z}\\psi_{n+1,\\mathbf{k}}$+H.c.\nIn terms of the parameterizations of Wang \\emph{et\nal.}\\cite{wang10}, we have\n\\begin{equation}\nh_{z}=-t_{z}(\\Gamma_{0}+i\\Gamma_{3}).\n\\end{equation}\nSince the $\\Gamma_{3}$ matrix now appears only in the $h_{z}$\npart of the coupled layers system, the difference between the\ntwo models enters only through the interlayer hopping term.\n\n\nNow introduce superconducting pairing and define the Nambu\nbasis as\n$\\phi_{n\\mathbf{k}}^{\\dagger}$=$[\\psi_{n\\mathbf{k}}^{\\dagger},\n\\psi_{n-\\mathbf{k}}^{T}]$. The intra-layer part of the\nBogoliubov de Gennes (BdG) Hamiltonian is then\n$\\hat{H}^{SC}_{\\parallel}$=$\\sum_{n\\mathbf{k}}\\phi_{n\\mathbf{k}}^{\\dagger}H_{SC}(\\mathbf{k})\\phi_{n\\mathbf{k}}$,\nin which\\cite{linder10}\n\\begin{equation}\nH_{SC}(\\mathbf{k})=\n\\begin{pmatrix} h_{0}(\\mathbf{k}) & \\underline{\\Delta}(\\mathbf{k}) \\\\\n-\\underline{\\Delta}^{\\ast}(-\\mathbf{k}) & -h^{\\ast}_{0}(-\\mathbf{k})\n\\end{pmatrix},\n\\end{equation}\nwhere $h_{0}(\\mathbf{k})$=$h_{xy}(\\mathbf{k})-\\mu\nI_{4\\times4}$, with $\\mu$ the chemical potential.\n$\\underline{\\Delta}(\\mathbf{k})$ is the 4$\\times$4 pairing\nmatrix. Ignoring the possibility of interlayer pairing, the\ninterlayer hopping terms are\n$\\hat{H}^{SC}_{\\perp}$=$\\sum_{n\\mathbf{k}}\\phi_{n\\mathbf{k}}^{\\dagger}H_{z}\\phi_{n+1,\\mathbf{k}}$+H.c.,\nin which\n\\begin{equation}\nH_{z}=\n\\begin{pmatrix} h_{z} & 0 \\\\\n0 & -h^{\\ast}_{z}\n\\end{pmatrix}.\n\\end{equation}\n\nOnce the pairing order is given, the surface spectral function\nis obtained from the retarded surface Green's functions, which\ncould be calculated in terms of standard transfer matrix\nmethod.\\cite{fastiter} In the simplest form of the method, the\n8$\\times$8 retarded surface Green's function\n$G(\\mathbf{k},\\omega)$ is obtained by self-consistent\ncalculation of $G(\\mathbf{k},\\omega)$ and a transfer matrix\n$T(\\mathbf{k},\\omega)$ as\\cite{fastiter,wang10}\n\\begin{subequations}\n\\begin{equation}\nG^{-1}=g^{-1}-H_{z}^{\\dagger}T,\n\\end{equation}\n\\begin{equation}\nT=GH_{z},\n\\end{equation}\n\\end{subequations}\nwhere $g$=$[zI_{8\\times8}-H_{SC}(\\mathbf{k})]^{-1}$\n($z$=$\\omega+i\\eta$) is the retarded Green's function for an\nisolated layer. $\\eta$ is the positive infinitesimal 0$^{+}$,\nwhich is replaced by a small positive number in realistic\ncalculations. Self-consistent calculation of the Green's\nfunction starts with $G$=$g$. The surface Green's function\ncould also be obtained in terms of other iteration schemes,\nsuch as the algorithm in Ref.\\cite{fastiter}. We have found no\ndifference between the results obtained in terms of different\niteration schemes.\n\nAfter the retarded Green's functions are at hand, the spectral\nfunction is obtained as\n\\begin{equation}\nA(\\mathbf{k},\\omega)=-\\sum_{i=1}^{4}\\text{Im} G_{ii}(\\mathbf{k},\\omega)\/\\pi.\n\\end{equation}\n\nSince we have now two orbital and two spin degrees of freedom,\nthere are many possible pairing channels for different possible\npairing mechanisms. Realistic theoretical determination of the\npairing symmetry requires the knowledge of pairing mechanism\nand reasonable parameter values, which are both lacking\npresently.\\cite{fu10} Here we will consider singlet and triplet\npairing orders as phenomenological input parameters. Their\nqualitative differences in spectral functions could help to\nidentify the pairing symmetry from experiments.\n\n\n\\subsection{gap opening in the topological surface states}\n\nBefore presenting the full spectral function, we first would\nlike to examine what happens to the topological surface states\ninherited from the normal state\\cite{wray10} upon the formation\nof a certain bulk pairing. The most salient feature of the\ntopological insulator is the presence of gapless surface states\n(3D) or edge states (2D).\\cite{kane05,bernevig06,fu07l,qi10pto}\nIn the case of Cu$_{x}$Bi$_{2}$Se$_{3}$, it is found that these\nsurface states in the non-superconducting Bi$_{2}$Se$_{3}$\npersist to the superconducting copper intercalated samples and\nare well separated from the bulk conduction band and hence\nwell-defined.\\cite{hor10,wray10} It is thus an interesting\nquestion what would happen to them if a certain pairing forms\nin the bulk. In this subsection, we give a simple criterion to\njudge whether a gap would be induced in the topological surface\nstates for an arbitrary bulk pairing.\n\nSuppose the chemical potential lies slightly above the bottom\nof the bulk conduction band where the topological surface\nstates are well separated and well defined.\\cite{wray10} If a\npairing is realized in the topological surface states, it\nshould occur between the two time reversal related states for\n$\\mathbf{k}$ and -$\\mathbf{k}$.\\cite{fu0809}\n\nWe first consider model (I). For our purpose, we would\nconcentrate on the positive energy branch of the topological\nsurface states. When the pairing occurs in the valence\nband\\cite{zhang10pa,zhang10pb}, the analysis and conclusion\nwould be similar. Since pairing occurs in the (k$_{x}$,\nk$_{y}$) space, we would ignore the $z$-dependence of the\nsurface modes when analyzing pairing properties. From the basis\nand the effective model obtained in Sec. II, the two\neigenvectors for a certain 2D wave vector are\n\\begin{equation}\n\\eta_{\\alpha}(\\mathbf{k})=\\frac{1}{\\sqrt{2}}[1, 0, \\alpha\\frac{k_{+}}{k}, 0]^{T},\n\\end{equation}\nwhere $\\alpha$ is `+' (`$-$') for the upper (lower) branch of\nthe surface states, $k_{\\pm}$=$k_{x}\\pm ik_{y}$,\n$k$=$\\sqrt{k_{x}^2+k_{y}^2}$. The annihilation operators of\nthese states are\n\\begin{equation}\nd_{\\mathbf{k}\\alpha}=\\frac{1}{\\sqrt{2}}[c_{1\\mathbf{k}\\uparrow}\n+\\frac{\\alpha k_{+}}{k}c_{1\\mathbf{k}\\downarrow}]\n\\end{equation}\nIf pairing is induced in the upper surface conduction band at\n$\\mathbf{k}$, the only possible pairing would be proportional\nto\n$d_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger}$.\\cite{fu0809}\nDenote the time-reversal operator as\n$\\mathcal{T}$.\\cite{fu0809,qi10b} Since\n$\\mathcal{T}c_{1\\mathbf{k}\\uparrow}^{\\dagger}\\mathcal{T}^{-1}$=$c_{1-\\mathbf{k}\\downarrow}^{\\dagger}$\nand\n$\\mathcal{T}c_{1\\mathbf{k}\\downarrow}^{\\dagger}\\mathcal{T}^{-1}$=$-c_{1-\\mathbf{k}\\uparrow}^{\\dagger}$,\nwe have $\\mathcal{T}\nd_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger}\\mathcal{T}^{-1}$\n=$\\frac{k_{+}^{2}}{k^2}d_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger}$.\nTo ensure the time-reversal symmetry of the pairing, the actual\npairing should be of the form\n\\begin{eqnarray}\n\\hat{\\Delta}^{I}_{SCB}(\\mathbf{k})&=&\\Delta_{0}\\frac{k_{+}}{k}\nd_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger} \\notag \\\\\n&=&\\frac{\\Delta_{0}}{2}[\\frac{k_{+}}{k}c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}\n-\\frac{k_{-}}{k}c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger} \\notag \\\\\n&&+(c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}\n-c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger})],\n\\end{eqnarray}\nwhere $\\Delta_{0}$ is the real pairing amplitude, which could\nbe an even or odd real function of $\\mathbf{k}$ depending on\nthe pairing realized in the bulk. `SCB' is abbreviation for the\nsurface conduction band (the topological surface states). Thus,\nthe surface conduction band only supports the anti-phase\n$p_{x}\\pm ip_{y}$ equal-spin triplet pairing and the\nspin-singlet pairing within orbital 1. No other bulk pairing\nchannels, especially those inter-orbital pairings, would open a\ngap in the topological surface states within the framework of\nmodel (I).\n\nFor model (II), the two eigenvectors of the surface states for\na certain 2D wave vector are (again, ignoring the\n$z$-dependence)\n\\begin{equation}\n\\eta_{\\alpha}(\\mathbf{k})=\\frac{1}{2}[1, -i, \\alpha\\frac{k_{+}}{k}, i\\alpha\\frac{k_{+}}{k}]^{T},\n\\end{equation}\nwhere $\\alpha$ is `+' (`$-$') for the upper (lower) branch of\nthe topological surface states. Following the same arguments as\nfor the first model, when the chemical potential cuts the\nupper branch of these well defined surface states the time\nreversal invariant pairing is of the form\n\\begin{eqnarray}\n\\hat{\\Delta}^{II}_{SCB}(\\mathbf{k})&=&\\Delta_{0}\\frac{k_{+}}{k}\nd_{\\mathbf{k}+}^{\\dagger}d_{-\\mathbf{k}+}^{\\dagger} \\notag \\\\\n&=&\\frac{\\Delta_{0}}{4}[\\frac{k_{+}}{k}(c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}\n-c_{2\\mathbf{k}\\uparrow}^{\\dagger}c_{2-\\mathbf{k}\\uparrow}^{\\dagger}) \\notag \\\\\n&&-\\frac{k_{-}}{k}(c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger}\n-c_{2\\mathbf{k}\\downarrow}^{\\dagger}c_{2-\\mathbf{k}\\downarrow}^{\\dagger}) \\notag \\\\\n&& +i\\frac{k_{+}}{k}(c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{2-\\mathbf{k}\\uparrow}^{\\dagger}\n+c_{2\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}) \\notag \\\\\n&& +i\\frac{k_{-}}{k}(c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{2-\\mathbf{k}\\downarrow}^{\\dagger}\n+c_{2\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger}) \\notag \\\\\n&& +(c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger}\n-c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger} \\notag \\\\\n&& +c_{2\\mathbf{k}\\downarrow}^{\\dagger}c_{2-\\mathbf{k}\\uparrow}^{\\dagger}\n-c_{2\\mathbf{k}\\uparrow}^{\\dagger}c_{2-\\mathbf{k}\\downarrow}^{\\dagger}) \\notag \\\\\n&& +i(c_{1\\mathbf{k}\\uparrow}^{\\dagger}c_{2-\\mathbf{k}\\downarrow}^{\\dagger}\n+c_{1\\mathbf{k}\\downarrow}^{\\dagger}c_{2-\\mathbf{k}\\uparrow}^{\\dagger} \\notag \\\\\n&& -c_{2\\mathbf{k}\\uparrow}^{\\dagger}c_{1-\\mathbf{k}\\downarrow}^{\\dagger}\n-c_{2\\mathbf{k}\\downarrow}^{\\dagger}c_{1-\\mathbf{k}\\uparrow}^{\\dagger})].\n\\end{eqnarray}\nAs in Eq. (17), $\\Delta_{0}$ could be a constant or a real\nfunction of $\\mathbf{k}$ compatible with the symmetry of one\npairing component contained in the above decomposition. Besides\nthe intra-orbital pairing channels active in the model (I),\nthere are two additional inter-orbital pairing channels that\nare effective in producing a gap in the topological surface\nstates. The last term in the above equation is just the\nodd-parity inter-orbital triplet pairing proposed by Fu and\nBerg\\cite{fu10} as a possible candidate of a topological\nsuperconductor to be realized in a superconductor like\nCu$_{x}$Bi$_{2}$Se$_{3}$.\n\nThe clear difference between $\\hat{\\Delta}^{I}_{SCB}$ and\n$\\hat{\\Delta}^{II}_{SCB}$ makes the distinction between model\n(I) and model (II) more obvious. Since the gap opening of the\nsurface states is measurable, it is highly desirable to\nascertain which model is the correct description of the\nunderlying physics of Bi$_{2}$Se$_{3}$ and Bi$_{2}$Te$_{3}$.\n\nPreviously, a simple effective model calculation indicates that\nno gap opens in the topological surface states for any triplet\npairing induced by proximity effect on the surface of a\ntopological insulator.\\cite{linder10} However, our analysis\nabove indicates that if the proximity induced triplet pairing\nis compatible with any of the triplet components explicit in\n$\\hat{\\Delta}^{I}_{SCB}$ ($\\hat{\\Delta}^{II}_{SCB}$) for model\nI (model II), then a full pairing gap could still be opened in\nthe topological surface states. Note that the real gap opening\npattern in the topological surface states also depends on\n$\\Delta_0$.\n\nExcept for the pairing channels explicit in\n$\\hat{\\Delta}^{I}_{SCB}$ for model (I) and\n$\\hat{\\Delta}^{II}_{SCB}$ for model (II), no other bulk pairing\ncould open a gap in the topological surface states. The\nexistence of surface Andreev bound states depends on whether or\nnot a gap opens in the topological surface states. We clarify\nthis matter in the next section.\n\n\n\\subsection{spectral function for typical pairing symmetries}\n\nObservation of superconductivity in Cu$_{x}$Bi$_{2}$Se$_{3}$\nbrings about anticipation that nontrivial topological\nsuperconducting states might be realized in this material. The\ntopological superconductor is defined as a state with a full\npairing gap in the bulk and nontrivial gapless Andreev bound\nstates on the surface.\\cite{fu10}\n\n\nPossible pairings realizable in a system depend on the symmetry\nof the system and the specific pairing mechanism. In the case\nof pairing induced by short range electron density-density\ninteractions, Fu and Berg identified four possible pairing\nchannels.\\cite{fu10} However, if the pairing is induced by more\nlong-range interactions, such as the electron-phonon\ninteraction, other pairing channels (e.g., in which the pairing\npotential is $\\mathbf{k}$ dependent) would also be possible. In\nthe following we would analyze several typical pairings and\ncompare results of the two different models. In each case,\nthere are three typical situations as regards to the position\nof the chemical potential $\\mu$: (1) $\\mu$ lies in the bulk\ngap; (2) $\\mu$ lies above but close to the bottom of the bulk\nconduction band, where the topological surface states are well\nseparated from the continuum bulk conduction band; (3) $\\mu$\nlies far above the bulk conduction band bottom, where the\nsurface states have merged into the continuum conduction band.\nWhile the latter two cases are relevant to the superconducting\nstate of Cu$_{x}$Bi$_{2}$Se$_{3}$\\cite{hor10,wray10}, the first\ncase could be regarded as mimicking the proximity effect from\nan external superconductor.\\cite{fu0809,sau10,linder10} In this\npaper we would focus on the latter two situations. When the\nchemical potential lies in the valence\nband\\cite{zhang10pa,zhang10pb}, the results should be\nqualitatively similar for the same type of bulk pairing.\n\nFollow Wang \\emph{et al.}\\cite{wang10}, the model parameters\nare taken as $t$=$t_{z}$=0.5, $m$=-0.7 in most cases. 0.7 is\nhalf of the bulk band gap. The width of the bulk conduction\nband at $k_{x}$=$k_{y}$=0 is $2(2t_{z}-|m|)$=0.6. The small\npositive number $\\eta$ in the Green's functions is taken as\n10$^{-4}$.\n\n\n\\emph{- even-parity intra-orbital singlet pairing} First, we\nstudy the simplest possible pairing denoted by\n$\\underline{\\Delta}(\\mathbf{k})$=$i\\Delta_{0}s_{2}\\otimes\\sigma_{0}$.\nSpectral functions for the two different models are the same\nfor this pairing, so only one is presented in Fig. 1. Here and\nin the following, the degree of darkness indicates the\nintensity of the spectrum. The continuum portions of spectrum\nare contributions from the bulk states, which have small finite\namplitudes on the surface. Henceforth, they would be called\nbulk conduction band for simplicity. The contributions from the\ntopological surface states are somewhat speckled because we\nhave taken a finite grid in the $(\\mathbf{k},\\omega)$ plane to\ncalculate the spectral function. When the grid points are taken\nto be very dense, contributions from the topological surface\nstates will also become smooth. To see the qualitative behavior\nmore clearly, a reasonably large pairing amplitude\n$\\Delta_{0}$=0.1 is considered.\\cite{wray10} The result is\nnearly identical in the $\\Gamma$K direction (along $k_{x}$\naxis) and the $\\Gamma$M direction (along $k_{y}$ axis) of the\n2D reduced Brillouin zone (BZ). The topological surface states\nof both two models open a gap, which are consistent with the\nanalysis of the previous subsection. Since no Andreev bound\nstate exists, this pairing is topologically trivial. The other\nintra-orbital singlet pairings with a $\\mathbf{k}$-dependent\n$\\Delta_{0}$, which is an even function of $\\mathbf{k}$ could\nalso be considered, such as the $d_{x^2-y^2}$-wave pairing. In\nthese cases, there would be line nodes along the nodal\ndirections of the pairing gap.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=10cm,angle=0]{Fig1.ps}\n\\caption{Spectral function for even-parity intra-orbital $s$-wave pairing,\nfor two typical parameter sets for which the topological surface states at the chemical potential\n(a) are well separated from the bulk conduction band\nand (b) merges into the bulk conduction band.\nThe two models give identical results for this pairing. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\n\\emph{- even-parity inter-orbital singlet pairing} Since there\nare now two orbits, another singlet pairing exists in the\ninter-orbital channel. The pairing matrix for the $s$-wave case\nis\n$\\underline{\\Delta}(\\mathbf{k})$=$i\\Delta_{0}s_{2}\\otimes\\sigma_{1}$.\nThe corresponding spectral functions presented in Fig. 2 for\nthis pairing are still identical for the two models. They\ndiffer from the spectra of the former intra-orbital pairing\nchannel in at least two aspects. First, no gap opens in the\ntopological surface states, which is in agreement with the\ncriterion proposed in the previous subsection. Second, though a\nfull gap also opens in the continuum part of the spectrum, it\nis not constant and shows some $\\mathbf{k}$ dependence. As\nshown in Fig. 2(b), the continuum part of the spectrum even\nnearly closes at some special wave vectors for certain\nparameters. Another interesting feature is the strong\nredistribution of spectral weight between the continuum\nconduction band and the topological surface states. Some weight\nin the bulk conduction band part of the surface spectrum above\nthe chemical potential is depleted and transferred to the\ntopological surface states below the chemical potential. This\nredistribution arises from the particle hole mixing induced by\nthe presence of bulk superconducting pairing. As would see\nbelow, a similar feature is present for each bulk pairing that\ndoes not open a gap in the topological surface states.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=15cm,angle=0]{Fig2.ps}\n\\caption{Spectral function for even-parity inter-orbital $s$-wave pairing,\nfor three sets of parameters for which at the chemical potential\n(a) the topological surface states are well separated from the bulk conduction band,\n(b) the topological surface states are almost merged into the bulk conduction band,\nand (c) the topological surface states are well merged into the bulk conduction band.\nThe two models give identical results for this pairing. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\\emph{- odd-parity inter-orbital triplet pairing} We now\nconsider the odd-parity inter-orbital triplet pairing channel,\nproposed by Fu and Berg as a candidate for possible nontrivial\ntopological superconducting states in\nCu$_{x}$Bi$_{2}$Se$_{3}$.\\cite{fu10} The pairing matrix is\n$\\underline{\\Delta}(\\mathbf{k})$=$\\Delta_{0}s_{1}\\otimes\\sigma_{2}$.\nAs was shown in Fig. 3, the spectral functions for the two\nmodels differ greatly. When the chemical potential is close to\nbottom of the bulk conduction band, the surface conduction band\nis still gapless for model (I) but opens a gap for model (II),\nin agreement with the analysis in the above subsection. Another\nessential difference is the existence or not of Andreev bound\nstates. For model (I), a band of Andreev bound states appears\ninside of the insulating gap of the continuum which\ncontinuously connects to the topological surface states. While\nfor model (II), there is a point node at (0, 0), no Andreev\nbound state exists inside of the gap region. When the chemical\npotential is increased to the position where the surface\nconduction band has almost merged into the continuum part of\nthe surface spectrum (corresponding to contribution from the\nbulk conduction band), surface spectra for the two models are\nas shown in Fig. 4. Since now there is no well separated\nsurface conduction band, a full gap opens also for the model\n(I). However, a band of Andreev bound states still exists. When\nwe further increase the chemical potential to $\\mu>1.3$ (for\n$m$=-0.7), there is no state close to the $\\Gamma$ point, then\nthere would be no Andreev bound state even for model (I). A\nstrong redistribution of spectral weight arising from the\nparticle hole mixing between the continuum bulk conduction band\nand the topological surface states is observed in results for\nmodel (I).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=10cm,angle=0]{Fig3.ps}\n\\caption{Spectral function for odd-parity inter-orbital triplet pairing\nin the opposite spin pairing channel, for\n(a) model (I) and (b) model (II). As shown are the cases for which\nthe topological surface states are well separated from the bulk conduction band at the chemical potential.\nThe parameters are as shown on the figures. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=10cm,angle=0]{Fig4.ps}\n\\caption{Spectral function for odd-parity inter-orbital triplet pairing\nin the opposite spin pairing channel, for\n(a) model (I) and (b) model (II). As shown are the cases for which\nthe topological surface states are merged into the bulk conduction band at the chemical potential.\nThe parameters are as shown on the figures. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\nBesides the pairing studied above, there are also other\nodd-parity pairings. Since they are possibly related with\nnontrivial topological superconducting phases, we analyze\nseveral of them in the following.\n\n\\emph{- odd-parity intra-orbital singlet pairing} In this case,\nevery orbit pairs into a spin-singlet, but the two orbits has a\nrelative $\\pi$ phase difference and is hence odd in\nparity.\\cite{fu10} The pairing matrix is denoted as\n$\\underline{\\Delta}(\\mathbf{k})$=$i\\Delta_{0}s_{2}\\otimes\\sigma_{3}$.\nThe corresponding spectral functions for the two models are\npresented in Figs. 5(a) and 5(b) for chemical potential close\nto the bottom of the bulk conduction band and thus the\ntopological surface state\n is well defined. A comparison with the previous\npairing, shown in Figs. 3 and 4, indicates that the results are\ninterchanged between the two models. Gap opening in the\ntopological surface states again follows the expectation from\nthe previous subsection. The Andreev bound states in the bulk\nband gap for the second model again connect continuously to the\nprotected topological surface states. The behaviors for higher\nchemical potentials, as shown in Figs. 5(c) and 5(d), are\nsimilar to that of the previous pairing shown in Fig. 4 with\nthe two models interchanged.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=8cm,angle=0]{Fig5.ps}\n\\caption{Spectral function for odd-parity intra-orbital $s$-wave pairing, for\nmodel (I) ((a) and (c)) and model (II) ((b) and (d)).\nFor (a) and (b) ((c) and (d)), the topological surface states\nare well separated from (merged into) the bulk conduction band\nat the chemical potential. Spectrum along other directions\n(crossing the $\\Gamma$ point) are qualitatively identical.}\n\\end{figure}\n\n\n\\emph{- odd-parity inter-orbital equal-spin triplet pairing}\nAnother interesting possibility is the equal-spin pairing\nchannel. Here we consider the two fold degenerate inter-orbital\npairing channels as proposed by Fu and Berg.\\cite{fu10} This\nkind of pairing could be favored by interorbital ferromagnetic\nHeisenberg interactions.\\cite{santos10} The two independent\nchoices for the pairing matrix are\n$\\underline{\\Delta}^{(1)}(\\mathbf{k})$=$i\\Delta_{0}s_{0}\\otimes\\sigma_{2}$\nand\n$\\underline{\\Delta}^{(2)}(\\mathbf{k})$=$\\Delta_{0}s_{3}\\otimes\\sigma_{2}$.\nFor these pairings, it is easy to see that no gap opens in the\ntopological surface states for both models. Since results for\nthe two models are identical, we only show those for the model\n(I). As shown in Fig. 6 for\n$\\underline{\\Delta}^{(1)}(\\mathbf{k})$, a peculiar anisotropic\nAndreev bound state structure is observed. An important\ndifference of this pairing from the above odd-parity pairing\nchannels is that it is anisotropic with respect to $k_{x}$ and\n$k_{y}$. Though the bulk dispersion is gapless in the\n$k_{y}k_{z}$ ($k_{x}k_{z}$) plane for\n$\\underline{\\Delta}^{(1)}(\\mathbf{k})$\n($\\underline{\\Delta}^{(2)}(\\mathbf{k})$), there is still a band\nof Andreev bound states for the wave vectors smaller than\n$k_{F}$ where a gap opens. The peculiar feature of the Andreev\nbound states along $k_{y}$ ($k_{x}$) for\n$\\underline{\\Delta}^{(1)}(\\mathbf{k})$\n($\\underline{\\Delta}^{(2)}(\\mathbf{k})$) is that they are\ndispersion-less (that is, completely flat).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm,height=8cm,angle=0]{Fig6.ps}\n\\caption{Spectral function for odd-parity inter-orbital triplet pairing\nin the equal spin pairing channel, for\nmodel (I) and $\\underline{\\Delta}^{(1)}(\\mathbf{k})$=$i\\Delta_{0}s_{0}\\otimes\\sigma_{2}$.\n(a) and (c) are along the $k_{x}$ direction while (b) and (d) are\nalong the $k_{y}$ direction.\nThe parameters are as shown on the figure.}\n\\end{figure}\n\nBesides the Andreev bound states within the gap, the\nredistribution of spectral weights arising from the particle\nhole mixing is also very interesting. An important feature is\nthe appearance of a linear band beyond the Fermi momentum and\nbelow the chemical potential, existing as a particle hole\nsymmetric band of the original topological surface states in\nthe normal phase. Once a bulk superconducting pairing forms in\nthe topological insulator itself (and not in the intercalated\ncopper) in Cu$_{x}$Bi$_{2}$Se$_{3}$, such a linear dispersive\nband is always there, no matter a gap opens or not in the\ntopological surface states. To see the above feature more\nclearly, we show in Figs. 7(a) and 7(b) the energy distribution\ncurves (EDC) for several typical wave vectors for two typical\npairings and parameter sets, which are same as those of Figs.\n1(a) and 3(a), respectively. The linear band mentioned above\nappears as a well defined peak slightly below the chemical\npotential. As the wave vector increases and shifts away from\nthe Fermi momentum, the peak deviates linearly from the\nchemical potential and the height and width of it both decrease\nrapidly, which is in agreement with the fact that\nsuperconducting pairing forms only close to the chemical\npotential. The integrated weight of the linearly dispersive\npeaks in Figs. 7(a) and 7(b) are shown in Fig. 7(c), as a\nfunction of the wave vector. If the gap in the superconductors\nrealized from a topological insulator is larger than what is\nreported in Ref. \\cite{wray10}, the above linear dispersive\nstructure in the EDC could be detectable by ARPES for the wave\nvectors close enough to the Fermi momentum.\\cite{hor10,wray10}\nThen this well defined peak structure arising from the\ntopological surface states could be used as a good indicator of\nthe formation of superconducting correlation in\nBi$_{2}$Se$_{3}$ and the involvement of the topological surface\nstates in the superconducting phase.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7cm,height=12cm,angle=0]{Fig7.ps}\n\\caption{EDC for three typical wave vectors for\n(a)even-parity intra-orbital singlet pairing\n(with parameters same as in Fig. 1(a)) and\n(b)odd-parity inter-orbital triplet pairing\n(with parameters same as in Fig. 3(a)) within model (I).\nThe parameters are as shown on the figures.\n(c)The integrated weight of the linearly dispersive peaks\nslightly below the chemical potential with parameters corresponding to (a) and (b),\nrespectively.}\n\\end{figure}\n\n\nFrom the above results for five different pairing symmetries,\nwe observe a simple rule for the existence of nontrivial\nsurface Andreev bound states. For odd-parity pairings, when a\nfull gap opens in the continuum part of the surface spectrum\nbut no gap opens in the topological surface states, a band of\nsurface Andreev bound states would arise which traverses the\nbulk pairing gap. This criterion is verified also by\ncalculations for other superconducting pairings not presented\nhere. The results in this subsection are summarized in Table I.\n\n\\begin{table}[ht]\n\\caption{Summary of results for the bulk pairings considered\nexplicitly in the present work. Results for two models, (I) and\n(II), are compared. `TSS' and `ABS' are the abbreviations for\n`topological surface states' and `Andreev bound states',\nrespectively. `$+$' and `$-$' means the even-parity and\nodd-parity pairings. For `Gap in TSS', `Y' and `N' represents\nthat a gap could and could not open in the topological surface\nstates. For `ABS', `Y' and `N' denotes that Andreev bound\nstates exist and do not exist on the surface for a certain bulk\npairing.} \\centering\n\\begin{tabular}{c c c c c c}\n\\hline\\hline\n$\\underline{\\Delta}$($\\mathbf{k}$) & $is_{2}\\otimes\\sigma_{0}$\n& $is_{2}\\otimes\\sigma_{1}$ & $s_{1}\\otimes\\sigma_{2}$\n& $is_{2}\\otimes\\sigma_{3}$ & $is_{0}\\otimes\\sigma_{2}$ \\\\ [0.5ex]\n\\hline\n$P$ & $+$ & $+$ & $-$ & $-$ & $-$ \\\\\nGap in TSS: (I) & Y & N & N & Y & N \\\\\nGap in TSS: (II) & Y & N & Y & N & N \\\\\nABS: (I) & N & N & Y & N & Y \\\\\nABS: (II) & N & N & N & Y & Y \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\subsection{the surface Andreev bound states}\n\nFor some superconducting pairings, such as the $p\\pm ip$ wave\npairing, it was known that Majorana fermions exist as gapless\nsurface or edge modes.\\cite{qi10b} According to an argument by\nLinder \\emph{et al}, all zero energy Andreev bound states\nemerging from the nondegenerate (that is, on each surface)\ntopological surface states should be Majorana\nfermions.\\cite{linder10} As regards our case, in the parameter\nregion where the topological surface states are well separated\nfrom the bulk conduction band, one observation is that the\nsurface Andreev bound states in the gap region connect\ncontinuously to the topological surface states inherited from\nthe normal phase (e.g., Fig. 3(a)). Since the topological\nsurface states are spin polarized helical and nondegenerate,\nthe surface Andreev bound states on each surface should also be\nnondegenerate. Then according to the arguments by Linder\n\\emph{et al}\\cite{linder10}, the zero energy Andreev bound\nstates presented in the above section should also be Majorana\nfermions.\n\n\nTo see more clearly the properties of the surface Andreev bound\nstates, we perform numerical calculations on a finite layer\nsuperconducting film. As an example, we will analyze the\nodd-parity inter-orbital triplet pairing channel described by\n$\\underline{\\Delta}(\\mathbf{k})$=$\\Delta_{0}s_{1}\\otimes\\sigma_{2}$,\nwithin model (I). Fig. 8(a) shows the dispersion for a fifty\nlayer film. It reproduces all the basic features in the\nspectral function (see Fig. 3(a)). Enlargement of the low\nenergy dispersion (Fig. 8(b)) shows that dispersion of the\nsurface Andreev bound states is linear close to the $\\Gamma$\npoint. For all the bulk pairings studied above, the dispersion\nfor the superconducting film reproduces well the features of\nthe corresponding spectral function. Figure 9(a) shows the wave\nfunction amplitudes for the surface state localized on the top\nseveral layers. The corresponding behavior for the topological\nsurface states in the normal phase is presented in Fig. 9(b).\nThe decay behavior of the surface bound states into the bulk in\nFig. 9(a) is seen to change continuously from oscillatory\nexponential decay in the gap region\\cite{fu10} to monotonic\nexponential decay outside the gap region. That is, the state\nchanges from particle-hole mixed superconducting quasiparticle\nto the topological surface states in the normal phase.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,height=10cm,angle=0]{Fig8.ps}\n\\caption{(a) Dispersion of a fifty layer superconductor emerging from model (I)\nand for the odd-parity inter-orbital triplet pairing $\\underline{\\Delta}(\\mathbf{k})$=$\\Delta_{0}s_{1}\\otimes\\sigma_{2}$.\nParameters used are $\\mu$=0.9, $\\Delta_{0}$=0.1, and\n$m$=-0.7. (b) An enlargement of the small wave vector and low energy part of (a).}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm,height=10cm,angle=0]{Fig9.ps}\n\\caption{(a) Decay of the wave function amplitude with layer number\nfor the surface Andreev bound states localized on the top several layers,\nresults are obtained for the superconducting state emerging from model (I)\nand for the odd-parity inter-orbital triplet pairing $\\underline{\\Delta}(\\mathbf{k})$=$\\Delta_{0}s_{1}\\otimes\\sigma_{2}$.\nThe parameters are $\\mu$=0.9, $\\Delta_{0}$=0.1, and\n$m$=-0.7. The Fermi wave vector in the $k_{x}$ direction is\nabout 0.19$\\pi$. (b) Decay of the wave function amplitude with\nlayer number for the topological surface states localized on\nthe top several layers, for the normal states of model (I) with\n$m$=-0.7. The two insets show enlargements of the small\namplitude regions of the two figures.}\n\\end{figure}\n\n\n\nIn this paper, we have always been discussing a homogeneous\nphase both in the bulk and on the surface. However, in the\npresence of exotic surface excitations as vortices, novel\nMajorana fermion modes may appear in the vortex core even for\nbulk pairings with no surface Andreev bound states. There have\nalready been many papers focusing on this possibility, which\nusually start from the effective\nmodel.\\cite{qi09l,santos10,hosur10}\n\n\nThe Andreev bound states that appear in many different\nsuperconducting pairings as shown above confirm the idea that\nsuperconducting states realized in a topological insulator are\nvery probable to have nontrivial topological characters. The\nAndreev bound states, if exist, should be easily detectable in\na tunneling type experiments as a well defined zero energy\npeak. Another way to detect the Majorana fermions as zero\nenergy Andreev bound states is to take advantage of the various\nphase sensitive transport devices proposed to produce and\nmanipulate the Majorana fermions.\\cite{fu0809,law09}\n\n\n\n\n\\section{summary}\nIn this paper, we have discussed the surface spectral function\nof superconductors realized from a topological insulator, such\nas the copper-intercalated Bi$_{2}$Se$_{3}$. These functions\nare calculated by projecting bulk states to the surface for two\ndifferent models used previously for the topological insulator.\nDependence of the surface spectra on the symmetry of the bulk\npairing order parameter are discussed with particular emphasis\non the odd-parity pairing. When an odd-parity pairing opens a\nfull gap in the bulk, but not for the topological surface\nstates, zero energy Andreev bound states are shown to appear on\nthe surface. When the topological surface states are well\nseparated from the bulk conduction band, the redistribution of\nspectral weight induced by the onset of superconductivity\nproduces a linearly dispersive peak structure beyond the Fermi\nmomentum and below the chemical potential. It is proposed as a\ncriterion for confirming that superconductivity occurs in the\nBi$_{2}$Se$_{3}$ (and not in copper) and the topological\nsurface states are involved in the superconducting phase. The\nzero energy surface Andreev bound states are argued to be\nMajorana fermions.\n\n\n\n\n\\begin{acknowledgments}\nWe thank Peter Thalmeier for helpful discussions. This work was\nsupported by the NSC Grant No. 98-2112-M-001-017-MY3. Part of\nthe calculations was performed in the National Center for\nHigh-Performance Computing in Taiwan.\n\\end{acknowledgments}\\index{}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nIt is known that the Horizontal Branch (HB) morphology in globular clusters of similar metallicities varies and the identification of a parameter, other than metallicity, responsible for this has originated much discussion in the astronomical literature. Recent findings that more massive clusters tend to\nhave HBs extended further into higher temperatures led Recio-Blanco et al. (2006) to suggest the cluster's total mass may be the \"second parameter\". However since the HB morphology varies\n due to stellar evolution, age is still a strong candidate for the\n second parameter status\n(e.g. Stetson et al. 1999; Catelan 2000 and references therein).\n\nRR Lyrae stars are outstanding stars in the HB whose absolute magnitudes and iron abundances have been\ncarefully calibrated (Clementini et al. 2003; Layden 1994) and therefore play an important role in studies of the structure of the Galaxy.\nThe estimation of individual physical parameters of RR Lyrae stars in globular clusters provides relevant insights not only into the distance and iron abundances of their parent clusters (e.g. Arellano Ferro et al. 2008a; 2008b), but also imposes constraints on the structure of the HB and on the stellar evolution in this phase (see e.g. Zinn, 1993; van den Bergh 1993; Zinn 1996).\n\nDetermination of the physical parameters in RR Lyrae stars can\nbe attained from the analysis of data obtained\nfrom fundamental techniques in astronomy such as photometry or\nspectroscopy. High dispersion spectroscopy produces accurate estimates\nfor some of the atmospheric physical parameters, i.e.,\ntemperature, surface gravity and metallicity, that have been used to calibrate\nlow resolution methods such as the $\\Delta S$ method (Preston 1959). However this approach is\nlimited mostly to brighter objects since the extended telescope time required\nto study fainter objects is very\ncompetitive and hence scarce. However,\nisolated examples of accurate spectroscopy of objects around $V\n\\sim 16-17$~mag do exist (e.g. James et al. 2004, Gratton et al. 2005,\nYong et al. 2005, Cohen et al. 2005).\n\n\nAlternative approaches to physical parameter estimations are\ntheoretical and semi-empirical calibrations of photometric indices,\nsuch as the synthetic color grids from atmospheric models (e.g.\nLester, Gray \\& Kurucz 1986) or, for RR Lyrae stars,\nthe decomposition of light curves in Fourier harmonics and the\ncalibration of the Fourier parameters in terms of key physical\nparameters (e.g. Simon \\& Clement 1993; Kov\\'acs 1998; Jurcsik 1998,\nMorgan et al. 2007). Kov\\'acs and his collaborators, in an extensive\nseries of papers (e.g. Kov\\'acs \\& Jurcsik 1996, 1997;\nJurcsik \\& Kov\\'acs 1996; Kov\\'acs \\& Walker, 2001), have\ndeveloped purely empirical relations from the Fourier analysis of\nthe light curves. The basic stellar parameters are based on the\nassumption that the period and the shape of the light curve are\ndirectly correlated with the physical parameters (or quantities\nrelated to them) such as the iron abundance [Fe\/H], absolute magnitude\n$M_V$ and effective temperature $T_{\\rm eff}$.\n\nIn this paper we shall use the Str\\\"omgren photometry of seven RR\nLyrae stars in Bootes to estimate their physical parameters.\nFurthermore, since we have obtained simultaneous $uvby-\\beta$\nphotoelectric photometry, the precise variation of the stellar\nbrightness and colors along the cycle of pulsation for\neach star can also be used to provide rigid constraints\non the physical parameters.\n\n\n\\section{Observational material and Reductions}\n\\label{sec:Observations}\n\n\nSome of the stars included in the present work were part of\nthe observing program in previous studies as secondary targets\n(Lampens et al. 1990 and Pe\\~na 2003). Other seasons were devoted\nentirely to some of these RR Lyrae stars. The log of the various seasons\nis given in Table \\ref{LOG}. The observations were obtained with the 1.5 m\ntelescope at the Observatorio Astron\\'omico Nacional of San Pedro\nM\\'artir, Mexico. The telescope was equipped with a six channel\nspectrophotometer described in detail by Schuster \\& Nissen (1988).\nAll the data reductions were performed using the NABAHOT\npackage (Arellano Ferro \\& Parrao 1989).\n\n\\begin{table*}[!ht]\n\\small{\n\\begin{center}\n\\caption[] {\\small Log of the observing seasons} \\hspace{0.01cm}\n \\label{LOG}\n\\begin{tabular}{lcc}\n\\hline \\hline\nStar & Initial date & Final date \\\\\n\\hline \\hline\nAE Boo & Jun 08, 2001 & Jun 11, 2001 \\\\\nTV Boo & Feb 04, 2002 & Feb 11, 2002 \\\\\nRS Boo \\& TV Boo & Apr 08, 2004 & Apr 11, 2004 \\\\\nFull sample& May 27, 2005&Jun 16, 2005\\\\\n\\hline \\hline\n\\end{tabular}\n\\end{center}\n}\n\\end{table*}\n\nIn those seasons in which the present RR Lyrae stars were supplementary\nobjects, fewer data\npoints were obtained each night.\nThe accumulated time span including all the seasons, however, was\nlarge enough to cover the whole cycle of all the stars in our sample.\nIn each season the same\nobserving routine was employed: a multiple series of integrations\nwas carried out, often five 10 s integrations of the star, to the\naverage of which a one 10 s integration of the sky was subtracted. A\nseries of standard stars taken from the Perry, Olsen \\& Crawford (1987)\nand Olsen(1983) was also observed with the same procedure\non each night to transform the data into the standard system. The\nseasonal transformation coefficients, except\nfor the 2002 season, are reported in Table 2. The indicated coefficients\nare those in the following equations (Gronbech, Olsen \\& Str\\\"omgren 1976).\n\n\\begin{eqnarray}\nV = A + B~(b-y)_{st} + y_{inst} \\\\\n(b-y)_{st} = C + D~(b-y)_{inst} \\\\\nm_{1~st} = E + F m_{1~inst} + G ~(b-y)_{inst} \\\\\nc_{1~st} = H + I m_{1~inst} + J ~(b-y)_{inst} \\\\\n\\beta_{1~st} = K + L~\\beta_{1~inst}\n\\end{eqnarray}\n\n For the 2002 season the transformation was made by filter instead of by\ncolor indices. In this season the relations between the standard and the instrumental\nvalues for each filter were the following:\n\n\\begin{equation}\nV=y_{st}= a + b~ y_{inst}\n\\end{equation}\n\\begin{equation}\nb_{st} =c + d~ b_{inst}\n\\end{equation}\n\\begin{equation}\nv_{st}= e + f~ v_{inst}\n\\end{equation}\n\\begin{equation}\nu_{st}= g + h~ u_{inst}\n\\end{equation}\n\n\\noindent\nThe coefficients are reported in the bottom part of Table 2.\nThe mean dispersions of the transformation equations are $0.028$,\n$0.016$, $0.015$, $0.010$ mag in $u, v, b$ and $y$ respectively. It\nwas after this that the color indices were calculated. In Fig. \\ref{fig1}\nthe instrumental $y$ magnitude and color transformations are shown.\nIn Fig. \\ref{fig2} the relationships between the instrumental and\nthe standard magnitudes in the 2002 season are shown.\n\nThe final $uvby-\\beta$ photometry is available in electronic form from the Centre de Donn\\'ees Astronomiques,\nStrasbourg, France. The $V$ light curves are shown in Fig. \\ref{fig3}.\nIn the following section a brief description of the light and color curves for each star is given.\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=8.cm,height=8.cm]{fig1.eps}\n\\caption{Transformations from the instrumental to the standard $uvby$ system for the magnitude and colors. See eqs 1-5.}\n \\label{fig1}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\includegraphics[width=8.cm,height=8.cm]{fig2.eps}\n\\caption{Transformations from the instrumental $uvby$ magnitudes to the standard system.\nSee eqs. 6-9.}\n \\label{fig2}\n\\end{figure}\n\n\n\\begin{figure*}[ht]\n\\includegraphics[width=15.cm,height=12.cm]{fig3.eps}\n\\caption{V light curves of the sample stars.}\n \\label{fig3}\n\\end{figure*}\n\n\\begin{table*}[!t]\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \\tablecols{9}\n \n\\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{Transformation coefficients\nof each season}\n\\label{COEFS}\n \\begin{tabular}{lccccccrccccr}\n\\hline\nseason &A & B &C & D &E & F & G & H & I &J &K & L \\\\\n\\hline\nJun, 01 &18.076& $+0.020$ &1.354 & $0.991$&$-1.437$ & $1.012$ & $-0.002$&$-0.623$ & $1.001$ & $0.148$&$--$ &$--$ \\\\\nApr, 04 &18.839& $-0.010$ &1.539 & $0.993$&$-1.231$ & $0.949$ & $0.024 $&$-0.429$ & $0.988$ & $0.068$&$-- $ &$ --$ \\\\\nMay, 05 &19.000& $-0.007$ &1.531 & $0.983$&$-1.309$ & $1.068$ & $0.020 $&$-0.315$ & $1.031$ & $0.153$&$2.626$ &$-1.348$ \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n\\begin{center}\n \\begin{tabular}{lcccccccc}\n & a & b & c & d & e & f & g & h \\\\\n\\hline\nFeb, 02 & 19.1565 &1.0054 & 20.9731 & 1.0236 & 21.0131 & 1.0101 & 20.5204 & 1.0025 \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\section{Notes on individual stars}\n\n\nDue to the fact that in most of the seasons the RR Lyrae stars were\nnot the primary target, the sampling of these stars was not\nhomogeneous. For example, AE Boo was observed for four\nnights in 2001 for a relatively short time on each night, although the\nphase coverage was almost complete. In the 2005 season, only a couple of\npoints per night were obtained for most RR Lyrae stars\nalthough the time span of the observations covered seventeen nights.\nFor AE Boo around 180 points in V were gathered.\n\nThe other amply observed star was TV Boo which was measured in 2002 (250\ndata points), in 2004 (5 points) and in 2005 (13 points).\nAccording to Wils et al. (2006)\nTV Boo shows a Blazkho period of 10 d. Given the time span of our\nobservations on this star and the uncertainties in the V magnitude\n(0.03 mag) we can neither verify nor contradict this assertion.\nWe call\nattention to the fact that the time of maximum light of this star\ndoes not coincide with a phase value of zero implying a secular\nvariation of the period. We did not pursue this possibility.\n\nThe last star which was\nobserved in more than one season was RS Boo. The data cover the\nfull cycle and the maximum light coincides with phase zero. Of the\nremaining stars, ST Boo, TW Boo, UU Boo and XX Boo,\nall observed in the 2005 season at the rate of a couple of points per night and\nfor seventeen nights, all\nshow a maximum light shifted relative to phase\nzero at least with the light elements reported in Table 3.\nFor each of these stars a sample of around thirty-five data points was gathered.\n\n\n\\begin{table*}[!t]\n\n\\small{\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{Ephemerides and Fourier coefficients for the sample stars.}\n\\label{COEF}\n \\begin{tabular}{lcccccccccccc}\n \\hline\nID & HJD & P & $A_0$ & $A_1$ & $A_2$ & $A_3$ & $A_4$ & $\\phi _{21}$ & $\\phi _{31}$ & $\\phi _{41}$ & N & Type \\\\\n& 2400000.0+ & (days) &$\\sigma_{A_0}$ &$\\sigma_{A_1}$ &$\\sigma_{A_2}$ &$\\sigma_{A_3}$ &$\\sigma_{A_4}$ & $\\sigma_{\\phi _{21}}$& $\\sigma_{\\phi _{31}}$& $\\sigma_{\\phi _{41}}$& \\\\\n\\hline\nRS Boo & 41770.4900 & 0.37733896 & 10.470 &0.447 &0.207 & 0.105& 0.084 &4.088 &1.881 &5.858 & 8 & ab\\\\\n & & & 0.014 &0.019 &0.018 & 0.020& 0.018 &0.129 &0.215 &0.287 & & \\\\\nST Boo & 19181.4860 & 0.62229069 & 11.044 &0.383 &0.194 & 0.133& 0.097 &3.979 &2.129 &6.112 & 4 & ab\\\\\n & & & 0.017 &0.023 &0.024 & 0.024& 0.024 &0.177 &0.268 &0.358 & & \\\\\nTW Boo & 26891.2680 & 0.53227315 & 11.309 &0.359 &0.138 & 0.069& 0.058 &3.782 &1.827 &5.438 & 4 & ab\\\\\n & & & 0.027 &0.037 &0.039 & 0.035& 0.039 &0.359 &0.703 &0.772 & & \\\\\nUU Boo & 36084.4100 & 0.45692050 & 12.251 &0.504 &0.203 &0.086 & 0.055 &3.905 &2.074 &6.078 & 4 & ab\\\\\n & & & 0.022 &0.030 &0.032 &0.035 & 0.035 &0.212 &0.431 &0.652 & & \\\\\nXX Boo & 29366.6460 & 0.58140160 & 11.914 &0.257 &0.114 &0.051 & 0.039 & 4.087 & 2.726& 6.189 & 4 & ab\\\\\n & & & 0.012 &0.018 &0.015 &0.017 & 0.019 & 0.201 & 0.387& 0.476 & & \\\\\nAE Boo & 30388.2500 & 0.31489240 & 10.646 &0.202 &0.027 &0.008 & 0.008 & 4.903& 3.722& 2.377& 6 & c \\\\\n & & & 0.004 &0.006 &0.006 &0.006 & 0.006 & 0.223& 0.730& 0.670& & \\\\\nTV Boo & 24609.5150 & 0.31255936 & 11.020 &0.282 &0.078 &0.020 & 0.015 & 3.915& 1.666 & 6.043 & 7 & c \\\\\n & & & 0.001 &0.001 &0.001 &0.001 & 0.001 & 0.017& 0.051 & 0.074 & & \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n}\n\\end{table*}\n\n\n\n\\section{Iron abundance estimation}\n\n\\subsection{{\\rm [Fe\/H]} from the Fourier light curve decomposition}\n\nAs a first approach, the V light curves were decomposed\nin their harmonics and the Fourier coefficients used\nto estimate the iron abundance via semiempirical calibrations.\nIn order to do that, the light curves were phased with periods and the epochs listed in\nTable \\ref{COEF}, which were adopted from Kholopov et al. (1985).\n\nGiven the ephemerides, the light curves in Fig. \\ref{fig3} were fitted by the equation:\n\n\\begin{equation}\n\\label{fourier}\n m(t)=A_0 + \\sum_{k=1}^N A_k \\cos(\\frac{2\\pi}{P}k\n (t-E)+\\phi_k)\\nonumber\n\\end{equation}\n\n\nwhere $k$ corresponds to the $k-th$ harmonic of amplitude $A_k$ and displacement $\\phi _k$.\n\nThe Fourier coefficients, defined as:\n\n\n\\begin{eqnarray}\n \\phi_{ij}& = & j\\phi_i - i\\phi_j \\nonumber \\\\\n R_{ij}& = & A_i \/ A_j \\nonumber\n\\end{eqnarray}\n\n\\noindent\nhave been calibrated in terms of physical parameters.\nThe Fourier coefficients corresponding to the solid curve in Fig. \\ref{fig3}\nand the number of harmonics used to produce the best possible fit,\nare given in Table \\ref{COEF}. The number of significant harmonics depends on the\ndispersion of the light curve. Only significant harmonics were retained; and their influence on [Fe\/H] estimated from the Fourier decomposition will be discussed at the end of this section.\n\nFor the RRab stars, [Fe\/H] was estimated using the calibration of\nJurcsik \\& Kov\\'acs (1996);\n\n\n\\begin{equation}\n\\label{JK}\n {\\rm [Fe\/H]}_{\\rm J} = -5.038 ~-~ 5.394~P ~+~ 1.345~\\phi^{(s)}_{31},\n\\end{equation}\n\n\\noindent\nThe standard deviation in the above equation is 0.14 dex. In eq. \\ref{JK}, the phase $\\phi^{(s)}_{31}$ is calculated from a sine series. To convert the cosine series based $\\phi^{(c)}_{jk}$ into the sine series $\\phi^{(s)}_{jk}$,\none can use ~$\\phi^{(s)}_{jk} = \\phi^{(c)}_{jk} - (j - k){\\pi \\over 2}$.\nThe metallicity [Fe\/H]$_{\\rm J}$ from eq. \\ref{JK} can be converted to the\nmetallicity scale of Zinn \\& West (1984) (ZW) via [Fe\/H]$_{\\rm J}$ = 1.43 [Fe\/H]$_{\\rm ZW}$ + 0.88 (Jurcsik 1995). The values of [Fe\/H]$_{\\rm ZW}$ are given in Table \\ref{FEHDM}.\n\n\\begin{table}[!t]\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \\tablecols{4}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{[Fe\/H]$_{ZW}$ for the sample of RR Lyraes. }\n\\label{FEHDM}\n \\begin{tabular}{lccc}\n \\hline\nID & [Fe\/H]$_{ZW}$ & $D_m$ & Type \\\\\n\\hline\nRS Boo& $-0.84$ & 4.1 & ab \\\\\nST Boo& $-1.53$ & 1.2 & ab \\\\\nTW Boo& $-1.47$ & 1.9 & ab \\\\\nUU Boo& $-0.95$ & 5.9 & ab \\\\\nXX Boo& $-0.81$ & 5.1 & ab \\\\\nAE Boo& $-1.30$ & & c \\\\\nTV Boo& $-2.04$ & & c \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\nBefore applying eq.~\\ref{JK} to the five RRab stars in our sample, we calculated the\n$compatibility ~ condition ~ parameter$ $D_m$ which, according to\nJurcsik \\& Kov\\'acs (1996) and Kov\\'acs \\& Kanbur (1998), should be smaller that 3.0.\nThe values of $D_m$ for the five RRab stars are given in Table \\ref{FEHDM}.\nIt is worth commenting that the $D_m$ parameter calculated for the RRab stars does not seem\nto correspond to the quality and\/or density of the light curve. For instance, from\nFig. \\ref{fig3} the curve of RS Boo is clearly the best and that of ST Boo is among the worst, however the corresponding $D_m$ values are 4.1 and 1.2 respectively, which is contrary to what would be expected. Thus we decided to relax this criterion a bit and apply the decomposition to the five RRab stars in our sample.\nWe shall discuss the uncertainties in [Fe\/H] due to the number of harmonics used to fit the data later in this section.\n\nFor the RRc stars we have calculated [Fe\/H] using the recent calibration of Morgan et al. (2007), which\nprovides [Fe\/H] in the ZW scale;\n\n\\begin{eqnarray}\n\\label{MORG}\n{\\rm [Fe\/H]}_{\\rm {ZW}} = 52.466 P^2 - 30.075 P + 0.131 (\\phi^{(c)}_{31})^2 \\nonumber \\\\\n+ 0.982 \\phi^{(c)}_{31} -4.198 \\phi^{(c)}_{31} P + 2.424\n\\end{eqnarray}\n\nThe results are listed in Table \\ref{FEHDM}.\n\nEq. \\ref{JK} is calibrated using the compilation of $\\Delta S$ values of Suntzeff et al. (1994)\ntransformed into [Fe\/H] and the spectroscopic values [Fe\/H]$_K$ of Layden (1994). On the other hand,\neq. \\ref{MORG} is calibrated using the [Fe\/H] values of Zinn \\& West (1984) and Zinn (1985) which are\nthe weighted average of iron abundances obtained from an assortment of methods, including the\n $\\Delta S$ and the Q$_{39}$ indices. Therefore the values of [Fe\/H] derived from the Fourier approach\nfor both RRab and RRc stars, are not completely independent from the values of [Fe\/H] estimated from $\\Delta S$, however\na discussion of $\\Delta S$ and [Fe\/H] values found in the literature for the stars in our sample is\nof interest and we present it in the following section.\n\nPerhaps the most relevant source of uncertainty in the physical parameters estimated from the Fourier decomposition approach is the quality and density of the light curve and hence its mathematical representation, alas, the number of\nharmonics needed to fit the observed data. For the four RRab stars ST Boo, TV Boo,\nUU Boo and XX Boo, whose light curves are scattered and\/or not very dense, the number of significant harmonics 3 or 4 is rather undistinguishable. It can be noted that the values of $T_{\\rm eff}$ and $M_V$\nvary 25-90~K and 0.01-0.07 mag. respectively, while the most striking variation\ntakes place in [Fe\/H] where variations on average were $\\pm 0.19$ dex. This value is marginally larger than the declared mean standard deviations of eqs.\n\\ref{JK} and \\ref{MORG} (0.14 mag.) Thus we estimate the uncertainties in the values of [Fe\/H]$_{ZW}$ of about $\\pm 0.17$ dex. We have retained the cases with the smaller standard deviation from the fit and with exclusively significant harmonics in Table \\ref{COEF}.\n\n\\subsection{{\\rm [Fe\/H]} from the $\\Delta S$ parameter}\n\nIt is a common practice to estimate the value of [Fe\/H] for RR Lyrae stars\nusing the $\\Delta S$ metallicity parameter; Preston's (1959) method. In an extensive study of\nthe $\\Delta S$ parameter on RR Lyrae variables in the Galactic halo, Suntzeff et al.\n(1994) provided average values of $\\Delta\nS$ with errors of $0.3$ units, which correspond to\n$0.05$ dex in [Fe\/H]. They reported $\\Delta S$ values for three\nstars of the present study: RS Boo, TW Boo and TV Bootes with values 1.36, 2.3 and 11.6. For RS Boo and TV Bootes\nthey give numerous estimations of $\\Delta S$, but since they correspond to similar phases, we have taken the averages and marked them with asterisks in Table \\ref{deltaS}. $\\Delta S$ values for ST Boo, TW Boo and XX Boo\nare found in the paper by Smith (1990), the average or individual values are given in Table \\ref{deltaS}.\nWe note here the discrepant values for TW Boo from Suntzeff et al.\n(1994) and Smith (1990) despite their corresponding to similar phases of 0.12 and 0.17 respectively. Also two discrepant values for XX Boo but for very different phases\nare found in Smith (1990). A value\nfor TV Boo is given by Liu \\& Janes (1990) as 12.1 which was measured near maximum light.\nA summary of the $\\Delta S$ values for our sample stars is given in Table \\ref{deltaS}.\n\n\\begin{table*}[!t]\n\\small{\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \\tablecols{7}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{$\\Delta S$ values for sample stars}\n\\label{deltaS}\n \\begin{tabular}{lcccccr}\n \\hline\nID & Type & Spectra & $\\Delta S$\\tabnotemark{1} & $\\Delta S$\\tabnotemark{2} &$\\Delta S$\\tabnotemark{3} & Px (mas)\\\\\n\\hline\nRS Boo& ab & A7-F5 & 1.36*$^m$ & & & $0.11\\pm 1.40$\\\\\nST Boo& ab & A7-F7 & & 8.9* & & $1.19\\pm 1.61$\\\\\nTW Boo& ab & F0-F8 & 2.3 & 6.2,8.3$^m$ & & $-0.28\\pm 1.63$ \\\\\nUU Boo& ab & & & & & \\\\\nXX Boo& ab & & & 7.0:,10.1: & & \\\\\nAE Boo& c & F2 & & & & $0.32\\pm 2.00$ \\\\\nTV Boo& c & A7-F2 & 11.6*$^m$ & & 12.1 & $-0.07\\pm 1.60$ \\\\\n\\hline\n \\tabnotetext{1}{Suntzeff et al. (1994)}\n \\tabnotetext{2}{Smith (1990)}\n \\tabnotetext{3}{Liu and Janes (1990)}\n \\tabnotetext{*}{average of multiple measurements }\n \\tabnotetext{m}{values from near minimum light}\n \\tabnotetext{:}{unknown phase}\n\\end{tabular}\n\\end{center}\n }\n\\end{table*}\n\n\\begin{table*}[!t]\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \\tablecols{8}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{Physical parameters from the calibrations for studied RR Lyrae stars. }\n\\label{FEMV}\n \\begin{tabular}{lccccccccc}\n \\hline\nID & [Fe\/H]$_{ZW}$ & [Fe\/H]$_{\\Delta S}$ &[Fe\/H]$_{K}$ & $T_{\\rm eff}$ &$M_V$ & BC &$\\log L\/L_{\\odot}$ & d (pc) & N \\\\\n\\hline\nRS Boo& $-0.84$ & $-0.81$ &$-0.32 \\pm 0.09$ & 6905&0.79 & 0.00& 1.58 & 862 & 8 \\\\\nST Boo& $-1.53$ & $-1.82$ &$-1.86 \\pm 0.14$ & 6454&0.48 & -0.05&1.73 & 1295 & 4 \\\\\nTW Boo& $-1.47$ & $-0.90,-1.46,-1.74$ &$-1.41 \\pm 0.09$ & 6581&0.58 & -0.05&1.69 & 1396 & 4 \\\\\nUU Boo& $-0.95$ & &$-1.92 \\pm 0.20$ & 6818&0.56 & -0.01&1.68 & 2284 & 4 \\\\\nXX Boo& $-0.81$ & $-1.56,-1.98$ & & 6675&0.62 & -0.03&1.67 & 1818 & 4 \\\\\nAE Boo& $-1.30$ & & & 7384&0.58 & 0.026&1.74 & 1032 & 6 \\\\\nTV Boo& $-2.04$ & $-2.18$ & & 7199&0.59 & 0.026&1.75 & 1218 & 7 \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n\\end{table*}\n\nWhile converting $\\Delta S$ into [Fe\/H] is a current practice, one should bear in mind that\nthe determination of $\\Delta S$ is subject to numerous sources of uncertainty, as have been\ndiscussed for example by Butler (1975), Smith (1986; 1990) and Suntzeff, et al. (1994),\ne.g. $\\Delta S$ estimated during the rising light can differ greatly from that\nestimated during the declining brightness. The correction to minimum phase is done using\nempirical curves on the $\\Delta S-SpT(H)$-plane with considerable dispersion and personal\njudgment in their final definition (Smith 1986; 1990), and the weighted mean finally reported value of\n$\\Delta S$ is often obtained from a few measurements randomly distributed along the pulsation\ncycle of the star. Some of the above considerations may explain the large differences in the existing\nvalues of $\\Delta S$ for a given star (see Table \\ref {deltaS}).\n\n\nBeside the above sources of uncertainty in $\\Delta S$, we have to note that the transformation of\nthis observational parameter into [Fe\/H] encountered serious problems: the\nhigh dispersion abundance determinations in RR Lyrae are old (e.g. Butler 1975,\nButler \\& Deeming 1979, Butler et al. 1982) while modern high signal-to-noise,\nhigh dispersion digital data, analyzed with modern synthetic codes are non-existent. According to Manduca (1981), conversion of $\\Delta S$ to [Fe\/H] has calibration problems for the metal-rich and metal-poor domains and in fact his theoretical calibration is not linear. We have chosen however,\nto use the more recent\nempirical linear transformation of Jurcsik (1995); ${\\rm [Fe\/H]}_J = -0.190 \\Delta S$ $-0.027$,\nwhich is valid for ${\\rm [Fe\/H]}_J$ between 0.0 and $-2.3$ dex and for field and cluster\nRR Lyraes, to transform the $\\Delta S$ parameter\ninto iron abundances on the ZW scale.\nWe have first used the $\\Delta S$ values in Table \\ref{deltaS} and the formula of Jurcsik (1995), to calculate ${\\rm [Fe\/H]}_J$, and then brought this into the ZW scale as discussed in $\\S$ 4.1; we have labeled these iron values as [Fe\/H]$_{\\Delta S}$ and they are listed in Table \\ref{FEMV}.\n\nFor the sake of comparison, in columns 2-4 of Table \\ref{FEMV} we have listed\nthe values of the Fourier estimations of the metallicity, [Fe\/H]$_{ZW}$, the values from\n$\\Delta S$, [Fe\/H]$_{\\Delta S}$ and the independent estimations from Layden (1994)\n[Fe\/H]$_{K}$. [Fe\/H]$_{K}$ values are based on the strength of the Ca II K line and are in the ZW scale.\n\nFor RS Boo [Fe\/H]$_{K}$ is rather discrepant but the agreement between [Fe\/H]$_{ZW}$ and [Fe\/H]$_{\\Delta S}$ is excellent.\nFor ST Boo and TW Boo the agreement between the three estimations is very good especially considering the implicit uncertainties in the values of $\\Delta S$.\nLarge discrepancies are noted for XX Boo and TV Boo. In the case of XX Boo one may argue that the coverage of the light curve can be considerably improved and hence the Fourier value could be expected to improve. However in the case of TV Boo, whose light curve is well defined and very dense, and whose\n$\\Delta S$ values are not scattered, we do not have an explanation handy.\nNevertheless, as an overall comparison of approaches to the determination of the iron abundance in RR Lyrae stars, and\nconsidering the uncertainties involved in the estimation of the $\\Delta S$ parameter discussed above and the fact that in the Fourier solution the full shape of the light curve is included, we consider the Fourier decomposition a more solid approach.\n\n\\subsection{The effective temperature $T_{\\rm eff}$}\n\nThe effective temperature can also be estimated from the Fourier coefficients. For the RRab stars we used the calibrations of Jurcsik (1998)\n\n\\begin{equation}\n\\label{teffab}\n log~T_{\\rm eff}= 3.9291 ~-~ 0.1112~(V - K)_o ~-~ 0.0032~[Fe\/H],\n\\end{equation}\n\n\\noindent\nwith\n\n$$ (V - K)_o= 1.585 ~+~ 1.257~P ~-~ 0.273~A_1 ~-~ 0.234~\\phi^{(s)}_{31} ~+~ $$\n\\begin{equation}\n\\label{color}\n~~~~~~~ ~+~ 0.062~\\phi^{(s)}_{41}.\n\\end{equation}\n\n\\noindent\nEq. \\ref{teffab}\nhas a standard deviation of 0.0018 (Jurcsik 1998), but the accuracy of $log~T_{\\rm eff}$ is mostly set by the color eq. \\ref{color}. The error estimate on $log~T_{\\rm eff}$ is 0.003 (Jurcsik 1998).\n\nFor the RRc stars we have used the calibration of Simon \\& Clement (1993);\n\n\\begin{equation}\n log T_{\\rm eff} = 3.7746 ~-~ 0.1452~log~P ~+~ 0.0056~\\phi^{(c)}_{31}.\n\\end{equation}\n\nThe values obtained from the above calibrations are reported in column 5\nof Table \\ref{FEMV}.\n\n\\section{Distance to the field RR Lyrae}\n\n\\subsection{$M_V$ from the Fourier parameters}\n\nAbsolute magnitudes of RR Lyrae can also be estimated from the Fourier parameters.\nFor the RRab one can use the calibration of Kov\\'acs \\& Walker (2001);\n\n\\begin{equation}\n\\label{KWab}\nM_V(K) = ~-1.876~log~P ~-1.158~A_1 ~+0.821~A_3 +K.\n\\end{equation}\n\n\\noindent\nThe standard deviations in the above equation is 0.04 mag.\nThe zero point of eq.~\\ref{KWab}, K=0.43, has been calculated by Kinman (2002) using the prototype star RR Lyrae as calibrator, adopting the absolute magnitude $M_V= 0.61 \\pm 0.10$ mag\nfor RR Lyrae, as derived by Benedict et al. (2002) using the star parallax measured by the HST.\nKinman (2002) finds his result to be consistent with the coefficients of the $M_V$-[Fe\/H] relationship given by Chaboyer (1999) and Cacciari \\& Clementini (2003). All these results are consistent with the distance modulus of the LMC of $18.5 \\pm 0.1$ (Freedman et al. 2001; van den Marel et al. 2002; Clementini et al. 2003). Catelan \\& Cort\\'es (2008) have argued that the prototype RR Lyr has an overluminosity due to evolution of 0.064 $\\pm$ 0.013 mag relative to HB RR Lyrae stars of similar metallicity. This would have to be taken into account if RR Lyr is used\nas a calibrator of the constant $K$ in eq. \\ref{JK}. Considering this, Arellano Ferro et al. (2008b) have estimated a new value of $K = 0.487$.\n\nFor the sake of homogeneity and better comparison with previous results on luminosities of RR Lyrae stars (e.g. Arellano Ferro et al. 2008a,b), in the following we have adopted $K=0.43$.\n\nFor the RRc stars, the calibration of Kov\\'acs (1998) was used;\n\n\\begin{equation}\n\\label{KWc}\nM_V(K) = ~-0.961~P ~-0.044~\\phi^{(s)}_{21} ~+4.447~A_4 + 1.261.\n\\end{equation}\n\nThe standard deviation in the above equation is 0.042 mag. In fact\nwe have propagated the errors, given in Table \\ref{COEF}, in the\namplitudes $A_1$ and $A_3$ in eq. \\ref{KWab} and in\n$\\phi^{(s)}_{21}$ and $\\phi^{(s)}_{41}$ in eq. \\ref{KWc} and found\nthat they produce\n uncertainties in $M_V$ $\\sim 0.04$ mag.\nCacciari et al. (2005) have pointed out that in order for the eq. \\ref{KWc} to surrender absolute magnitudes in agreement with the mean magnitude for the RR Lyrae stars in the LMC, $V_0 = 19.064 \\pm 0.064$ (Clementini et al. 2003), the zero point\nof the above equation should be decreased by 0.2$\\pm$0.02 mag.\nAfter this correction, we found the $M_V$ values for the RRc stars\nAE Boo and TV Boo reported in Table \\ref{FEMV}. These values of $M_V$ have been converted into\nlog $L\/L_{\\odot}$ (col. 5). The bolometric corrections for the average temperatures of RRab and RRc stars given in column 7 of Table \\ref{FEMV}, were estimated from the\n$T_{\\rm eff}$-BC$_V$ from the models of Castelli (1999) as tabulated in Table 4 of Cacciari et al. (2005).\n\n\\begin{table*}[!t]\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \\tablecols{8}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{$uvby-\\beta \\ $ photoelectric photometry and reddening of $\\delta$ Scuti stars in Bootes}\n\\label{delred}\n \\begin{tabular}{lccccccc}\n \\hline\nstar & $V$ & $b-y$ & $m_1$ & $c_1$ & $\\beta$ & $E(b-y)$ & $uvby-\\beta$ \\\\\n & & & & & & & source \\\\\n\\hline\n$\\iota$ Boo & 4.75 & 0.128 & 0.198 & 0.834 & 2.817 & 0.001 & 1\\\\\n$\\gamma$ Boo & 3.03 & 0.116 & 0.191 & 1.008 & 2.817 & 0.006& 1\\\\\n$\\kappa$$^2$Boo & 4.54 & 0.125 & 0.187 & 0.951& 2.806 & 0.001 & 1 \\\\\nCN Boo & 5.98 & 0.162 & 0.201 & 0.748 & 2.770 & 0.000 & 1\\\\\nYZ Boo & 10.57 & 0.151 & 0.178 & 0.868 & 2.768 & 0.000 & 1\\\\\nYZ Boo & 10.466 & 0.184 & 0.136 & 0.681 & 2.723 & 0.000 & 2 \\\\\n\\hline\n \\tabnotetext{1}{Rodr\\'{\\i}guez et al. (1994)}\n \\tabnotetext{2}{Pe\\~na et al. (1999)}\n \\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\subsection{Reddening}\n\nIn order to determine distance and detailed variations of\nthe physical parameters along the cycle of pulsation of each star,\nit is necessary to first estimate the reddening. For\nfield stars however, a proper determination of reddening is complex\nand no direct method is known to us. In view of this, we have\ndetermined the reddening of different objects in the same direction of the sky.\nWe have chosen five $\\delta$ Scuti stars and the globular cluster NGC~5466 in Bootes\nand M3 which is very near NGC~5466, as indicators of reddening in that direction of the sky.\nDespite the large distance to M3 and NGC~5466 (10.4 and 15.9 kpc respectively),\n$E(b-y)\\sim 0.0$ as estimated from the $E(B-V)$ values listed by Harris (1996).\nIn the case of the $\\delta$ Scuti stars we have used the expressions derived by\nCrawford (1975, 1979) for F- and A-type stars respectively, with the zero point correction\nsuggested by Nissen (1988), to estimate the intrinsic color $(b-y)_o$ for\nstars near the main sequence. Two sources of $uvby-\\beta$ were considered;\nRodr\\1guez et al. (1994) and Pe\\~na et al. (1999). These $\\delta$ Scuti stars,\ntheir magnitude-weighted mean colors and $E(b-y)$ are listed in Table \\ref{delred}.\n>From these results it seems reasonable to conclude that the\nreddening for the sample RR Lyrae stars in Bootes is negligible and we shall assume a value of\nzero for all the stars in our sample.\n\n\\begin{figure}[t]\n\\includegraphics[width=8.cm,height=8.cm]{fig4.eps}\n\\caption{$M_v- {\\rm [Fe\/H]}$ relation derived from cluster RR Lyrae stars (Arellano Ferro et al. 2008b). Blue points are the five RRab stars in our sample. When available, two positions are given; the larger points are the finally adopted solution while the smaller points correspond to similar solutions with smaller number of harmonics. Open blue circles correspond to the two RRc stars in our sample.\nThe uncertainties in the values of the blue symbols are discussed in the text and indicated by the\nblue error bars.}\n \\label{fig4}\n\\end{figure}\n\n\\subsection{Distances}\n\nThe implied distances given by the Fourier absolute magnitudes, and the reddenings and bolometric\ncorrections discussed in $\\S$ 5.1 and 5.2 are listed in column 9 of Table \\ref{FEMV}. These\ndistances can be compared with estimations from the parallaxes,\nfor those stars which have parallaxes determined by the new reductions of the Hipparcos data by\nvan Leeuwen (2007). For the stars\nRS Boo, ST Boo, TW Boo, AE Boo and TV Boo the parallaxes and their errors are listed in Table \\ref{deltaS}.\nIt should be noted that their errors are very large and hence the parallaxes for these stars very uncertain.\nExcept for ST Boo, those numerical values lead to distances much larger than the values derived from\nthe Fourier approach in Table \\ref{FEMV}, and if these distances are used to calculate the\ncorresponding absolute magnitude they lead to absurd results.\n\n\n\\begin{figure*}[t]\n\\includegraphics[width=16.cm,height=16.cm]{fig5.eps}\n\\caption{Model grids on the $(b-y)_0 - c_0$ plane along the stellar variations through the cycle.}\n \\label{fig5}\n\\end{figure*}\n\n\n\\section{$M_v$ from the Str\\\"omgren $c_1$ index}\n\nThe Str\\\"omgren $c_1$ index is a gravity (hence luminosity)\nsensitive parameter for late-A to F type stars (Str\\\"omgren 1966), and therefore\nit is thought to be useful in\ndetermining the absolute magnitude $M_V$ for RR Lyrae stars if properly\ncalibrated. In fact, in a recent paper Cort\\'es \\& Catelan (2008) have offered such calibration for RR Lyrae stars.\nThese authors have employed the\npseudocolour $c_0 \\equiv (u-v)_0 -(v-b)_0$, the fundamental period $P$\nand the metallicity $Z$ for a large number of synthetic RR Lyrae stars\nto calibrate $M_y$ and the colors $b-y$, $v-u$ and $u-v$ (their eq. 1).\n\n\\begin{table*}\n\\small{\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \\tablecols{7}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{Comparison of $M_V(F)$ obtained from the Fourier approach with two\ntheoretical calibrations from Cort\\'es \\& Catelan (2008).}\n\\label{MVC0}\n \\begin{tabular}{lcccccc}\n \\hline\n Star & [Fe\/H]$_{ZW}$ & $(c_1)$ & $M_y(c1)$\\tabnotemark{1} & $M_y(Z)$\\tabnotemark{2} & $M_V(F)$\\tabnotemark{3}& $M_V(F)$-$M_y(c1)$ \\\\\n \\hline\nRS Boo & $-0.84$ & ~~0.928 &~~0.884 & ~~0.765 & 0.79 & $-0.093$\\\\\n & & $\\pm 0.012$ & $\\pm 0.016$ & $\\pm 0.063$ & & \\\\\nST Boo & $-1.53$ & ~~0.922 &~~0.319 & ~~0.526 & 0.48 & $+0.163$\\\\\n & & $\\pm 0.010$ & $\\pm 0.013$ & $\\pm 0.030$ & & \\\\\nTW Boo & $-1.47$ & ~~0.893 &~~0.537 & ~~0.577 & 0.58 & $+0.045$\\\\\n & & $\\pm 0.017$ & $\\pm 0.022$ & $\\pm 0.039$ & & \\\\\nUU Boo & $-0.95$ & ~~0.941 &~~0.659 & ~~0.665 & 0.56 & $-0.098$\\\\\n & & $\\pm 0.017$ & $\\pm 0.023$ & $\\pm 0.051$ & & \\\\\nXX Boo & $-0.81$ & ~~0.814 &~~0.678 & ~~0.776 & 0.62 & $-0.057$\\\\\n & & $\\pm 0.011$ & $\\pm 0.018$ & $\\pm 0.063$ & & \\\\\nAE Boo & $-1.30$ & ~~1.002 &~~0.880 & ~~0.619 & 0.58 & $-0.298$\\\\\n & & $\\pm 0.003$ & $\\pm 0.004$ & $\\pm 0.044$ & & \\\\\nTV Boo & $-2.04$ & ~~1.133 &~~0.625 & ~~0.481 & 0.59 & $-0.034$\\\\\n & & $\\pm 0.006$ & $\\pm 0.006$ & $\\pm 0.018$ & & \\\\\n\\hline\n \\tabnotetext{1}{from $C_0, Z ~and~ P$ through eq. 1 of Cort\\'es \\& Catelan (2008).}\n \\tabnotetext{2}{from $Z$ through eq. 5 of Cort\\'es \\& Catelan (2008).}\n \\tabnotetext{3}{from the Fourier decomposition approach in this work, Table \\ref {FEMV}.}\n \\end{tabular}\n\\end{center}\n}\n\\end{table*}\n\nWe have fitted the $c_1$ curves with a curve of the form of eq.\n\\ref{fourier} to estimate the magnitude-weighted means $(c_1)$ given\nin column 3 of Table \\ref{MVC0}. These magnitude-weighted means\ndiffer from the intensity weighted means $$ by about 0.01 mag.\n(Catelan \\& Cort\\'es, 2008), which are smaller than the uncertainties for\n$c_1$ of our observations as derived from the standard stars (see\nTable 3 in Pe\\~na et al., 2007). Since $E(b-y) = 0.$ for the sample\nstars (sec. 5.2) we have taken $c_0 = c_1$. For the conversion of\n[Fe\/H] to Z we have used log Z = [M\/H] - 1.765; M\/H = [Fe\/H] + log\n(0.638 $f$ + 0.362) and $f = 10^{[\\alpha \/Fe]}$ (Salaris et al.\n1993). We adopted $\\alpha = 0.31$ as an appropriate value for halo\npopulation stars. The predicted values of the absolute magnitude\nfrom $c_1$, $M_y(c1)$, are given in column 4 of Table \\ref{MVC0}.\n\nIn their equation 5 Cort\\'es \\& Catelan (2008) have calculated a\nquadratic calibration of the $M_y$ - log Z relationship which can be\nused to calculate $M_y(Z)$ given in column 5 of Table \\ref{MVC0}. As\nargued by Cort\\'es \\& Catelan (2008) $M_y(c1)$ refers to the\nmagnitude of an individual star whereas $M_y(Z)$ is the absolute\nmagnitude of a star of similar metallicity. Thus we shall compare\nthe Fourier approach results of each star, $M_V(F)$, with their\ncorresponding $M_y(c1)$. This comparison is made in column 7. The\nuncertainties of $M_y(c1)$ are the resutl of propagating the\nuncertainties in $$ throught the equation 1 of Cort\\'es \\&\nCatelan (2008). The uncertainties $M_y(Z)$ are estimated by\npropagating the uncertinty in $\\sigma _{[Fe\/H]} = 0.17$ dex ($\\S$\n4.1). One can see that the dispersion in the calibration of equation\n1 in Cort\\'es \\& Catelan (2008) is $\\leq$ 0.01 mag (their Fig. 3)\nand we recall that the uncertainties in $M_V(F)$ is $\\pm 0.04$ mag\n($\\S$ 5.1). Therefore the differences in column 7 seem to be a bit\non the large side. It should be noted, however, that no systematics\ncan be seen and that if the $M_y(c1)$ values are plotted in Fig.\n\\ref{fig4} the dispersion of the field stars about the cluster $M_V-\n{\\rm [Fe\/H]}$ relationship becomes very large. Also it can be seen\nthat the values of $M_y(c1)$ for the two RRc stars, AE Boo and TV\nBoo are the most discordant despite of having the most densely\ncovered light curves and hence the smallest uncertainties.\n\n\n\\section{$M_V - {\\rm [Fe\/H]}$ relation}\n\nA recent linear version of the $M_V- {\\rm [Fe\/H]}$ relationship for RR Lyrae stars has been calculated by\nArellano Ferro et al. (2008a) based on the light curve decomposition technique of RR Lyrae stars in a\ngroup of globular clusters with a large range of metallicities. This relationship, reproduced in Fig \\ref{fig4}, has been amply discussed by Arellano Ferro et al. (2008b) who found it to be consistent with independent\nempirical linear versions and with theoretical non-linear versions after evolution from the Red Giant Branch is taken into account.\nTo check the consistency of our present results for the field stars in Bootes with those of RR Lyrae stars in globular clusters, we have plotted in Fig \\ref{fig4} the seven stars in our sample using the\nFourier [Fe\/H]$_{ZW}$ and $M_V$ reported in Table \\ref{FEMV}. It can be seen that with some larger scatter the Bootes stars distribution, given the uncertainties, follow the trend of the globular cluster RR Lyrae. The error bars correspond to the uncertainties in the Fourier calibrations of [Fe\/H]$_{ZW}$ and $M_V$, i.e., those of eqs. \\ref {JK}, \\ref {MORG}, \\ref {KWab} and \\ref {KWc}.\n\n\\begin{table*}\n\\small{\n\\begin{center}\n\\setlength{\\tabnotewidth}{\\columnwidth}\n \\tablecols{14}\n \n \\setlength{\\tabcolsep}{1.0\\tabcolsep}\n \\caption{$T_{\\rm eff}$ and $\\log~g$ variation ranges from the theoretical grids of LGK86}\n\\label{LGK86}\n \\begin{tabular}{lccccccccc}\n \\hline\nstar & [Fe\/H] & $T_{\\rm eff}$& $T_{\\rm eff}$ & $T_{\\rm eff}$ &\n$\\Delta T_{{\\rm eff}}$ & $\\log g$ &\n$\\log g$ & $\\Delta\\log g$ & $\\Delta V$ \\\\\n & adopted & Fourier & min & max & $K$ & min & max & dex & mag \\\\\n\\hline\nRS Boo & -1.0 & 6569 & 5700 & 8000 & 2300 & 2.2 & 3.5 & 1.2 & 0.82 \\\\\nST Boo & -1.5 & 6141 & 5700 & 7500 & 1800 & 1.5 & 3.0 & 1.5 & 0.73 \\\\\nTW Boo & -2.0 & 6262 & 5500 & 7000 & 1500 & 1.5 & 3.0 & 1.5 & 0.85 \\\\\nUU Boo & -1.0 & 6486 & 5500 & 8000 & 2500 & 1.5 & 3.0 & 1.5 & 0.63 \\\\\nXX Boo & -1.0 & 6350 & 5000 & 6800 & 1300 & 1.5 & 3.0 & 1.5 & 0.77 \\\\\nAE Boo & -1.5 & 7384 & 6100 & 7200 & 1100 & 1.9 & 2.8 & 0.9 & 0.48 \\\\\nTV Boo & -1.0 & 7199 & 6000 & 7500 & 1500 & 1.3 & 2.5 & 1.2 & 0.79 \\\\\n\\hline\n \\end{tabular}\n\\end{center}\n}\n\\end{table*}\n\n\\section{Determination of physical parameters along the pulsational cycle}\nGiven the\nsimultaneity in the acquisition of the data in the different color\nindices, once the reddening\nhas been inferred, it is possible to determine the variation of the\nphysical parameters of the star along the cycle.\nThis can be accomplished with the models developed\nparticularly for $uvby-\\beta \\ $ photometry by Lester, Gray \\& Kurucz\n(1986) (LGK86). The models have been built taking into account that the\n$uvby-\\beta \\ $ system is well designed to measure key spectral signatures\nthat can be used to determine basic stellar parameters. The\ntheoretical calibrations have the advantage of relating the\nphotometric indices to the effective temperature, surface gravity,\nand metallicity. LGK86 provide grids, on the plane $(b-y) - c_1$,\nof constant $T_{\\rm eff}$ and log g, for a large range of [Fe\/H] values.\nBased on the Fourier value [Fe\/H]$_{ZW}$, a model with the nearest [Fe\/H] value\nwas chosen for each star.\nIn Fig. \\ref{fig5} the cycle variation of each star on its corresponding grid is illustrated.\nThis allows an estimation of the $T_{\\rm eff}$ and log g variation ranges during the pulsation\ncycle and a comparison with the estimated temperature from the Fourier approach.\n\nIt is interesting to note that the studied stars have different\neffective temperatures and surface gravity limits, as well as\ndifferent ranges which cannot be determined with detail with only\nthe Fourier techniques. These results have been summarized in Table \\ref{LGK86}\nin which we have also included those determined through the\nempirical calibrations from the Fourier coefficients. As can be\nseen, both methods give analogous results. In Table \\ref{LGK86}\nthe variation ranges in V, $T_{\\rm eff}$ and log g are also indicated.\n\n\n\\section{Conclusions}\n\n>From data acquired in several photometric campaigns we have obtained\nextended $uvby-\\beta$ photometry over a relatively large time span\nfor two stars and data that adequately cover the cycle of pulsation\nfor all of them. The $V$ light curves have been Fourier decomposed\nand the corresponding Fourier parameters from their harmonics were\nused to calculate the iron abundance and luminosity of each star.\nThe reddening was estimated by considering different objects in\nthe same direction. The unreddened Str\\\"omgren indices $c_0$ and\n$(b-y)_0$ served to determine the variation along the cycle of\nthe effective temperature and surface gravity.\n\n The iron abundance [Fe\/H]$_{ZW}$ was calculated\nfirst from the Fourier decomposition of the light curve and the\ncalibration proposed by Kov\\'acs and co-workers and described with\ndetail in $\\S$ 4.1. We also utilized the $\\Delta S$\nparameter to estimate the metallicity. In $\\S$ 4.2 we amply discuss\n the uncertainties and the limitation of this technique. We\n conclude that the Fourier decomposition approach gives more\nreliable results.\n\nSince RR Lyrae stars are distance indicators, the individual estimations\nof the absolute magnitude for field stars from independent methods is of interest.\nThe absolute magnitude, $M_V (F)$, predicted from Fourier decomposition ($\\S$ 5.1),\nis reported with other two determinations.\nOnce the iron abundance and the reddening were determined, an\nindependent estimate of $M_V (c_1)$ can be made from the $c_1$ index and the pseudocolor\n$C_0$ (Cort\\'es \\& Catelan 2008). This is described in $\\S$ 6. Also\nin that section\nwe calculated $M_V (Z)$ from the metallicity alone making use of a theoretical\nquadratic $M_V - Z$ relationship offered by Cort\\'es \\& Catelan (2008).\nAll these\nresults were compiled in Table \\ref{MVC0}. It was pointed out that, although in some individual cases the differences between $M_V (F)$ and for example $M_V (c_1)$ were larger than the\nexpected from the uncertainties of the methods involved,\nno systematic trends could be seen and in some cases the agreement is fairly good.\nThe agreement does not seem to be related to the quality or the density of the\n$V$ light curve, since the disagreement is largest for the two RRc stars which also\nhave the best light curves in our sample.\n\nThe $M_V- {\\rm [Fe\/H]}$ relationship for RR Lyrae stars has been amply discussed in the\nliterature. A recent version of it calculated exclusively from the Fourier\ndecomposition approach of RRab and RRc\nstars in globluar clusters (Arellano Ferro et al. 2008b) was used to confront our\npresent results for the field stars (Fig \\ref{fig4}).\nIt can be seen that with some larger\nscatter the distribution of the Bootes stars follow the same trend of the\nglobular cluster RR Lyrae stars and it was noted that if the alternative results\nfor the absolute magnitude, $M_V (c_1)$ or $M_V (Z)$ were used the quality of the comparison would remain.\n\nWith $M_V (F)$ and the reddening we have reported the resulting distances for the\nsample stars (Table \\ref{MVC0}).\nIt would be desirable to compare these distances with the derived distances from\nan independent technique. Unfortunately, those determined from the\nnew reductions of the Hipparcos catalogue (van Leeuwen 2007) have such large errors that the comparison is impossible.\n\n\\vspace{0.5cm}\n\nAcknowledgements. We would like to thank the following people for\ntheir assistance: the staff of the OAN, M. S\\'anchez and V. Alonso\nat the telescope during some of the 2005 observations and A. Pani\nfor those of 2002, and L. Parrao for the reduction of the 2002\nseason. We are grateful to an anonymous referee for useful\ncorrections and suggestions. This paper was partially supported by\nPAPIIT-UNAM IN108106 and IN114309. J. Miller and J. Orta did the\nproofreading and typing, respectively. This article has made use of\nthe SIMBAD database operated at CDS, Strasbourg, France and ADS,\nNASA Astrophysics Data Systems hosted by Harvard- Smithsonian Center\nfor Astrophysics.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}