\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{ Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 160, pp. 1--25. \newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/160\hfil Vertical blow ups of capillary surfaces] {Vertical blow ups of capillary surfaces in $\mathbb{R}^3$, Part 2: Nonconvex corners} \author[T. Jeffres, K. Lancaster\hfil EJDE-2008/160\hfilneg] {Thalia Jeffres, Kirk Lancaster} % in alphabetical order \address{Thalia Jeffres \newline Department of Mathematics and Statistics \\ Wichita State University \\ Wichita, Kansas, 67260-0033, USA} \email{jeffres@math.wichita.edu} \address{Kirk Lancaster \newline Department of Mathematics and Statistics \\ Wichita State University \\ Wichita, Kansas, 67260-0033, USA} \email{lancaster@math.wichita.edu} \thanks{Submitted August 3, 2007. Published December 9, 2008.} \subjclass[2000]{49Q20, 53A10, 76B45} \keywords{Blow-up sets; capillary surface; Concus-Finn conjecture} \begin{abstract} The goal of this note is to continue the investigation started in Part One of the structure of ``blown up'' sets of the form $\mathcal{P}\times \mathbb{R}$ and $\mathcal{N}\times \mathbb{R}$ when $\mathcal{P}, \mathcal{N} \subset \mathbb{R}^{2}$ and $\mathcal{P}$ (or $\mathcal{N}$) minimizes an appropriate functional and the domain has a nonconvex corner. Sets like $\mathcal{P}\times \mathbb{R}$ can be the limits of the blow ups of subgraphs of solutions of capillary surface or other prescribed mean curvature problems, for example. Danzhu Shi recently proved that in a wedge domain $\Omega$ whose boundary has a nonconvex corner at a point $O$ and assuming the correctness of the Concus-Finn Conjecture for contact angles $0$ and $\pi$, a capillary surface in positive gravity in $\Omega\times\mathbb{R}$ must be discontinuous under certain conditions. As an application, we extend the conclusion of Shi's Theorem to the case where the prescribed mean curvature is zero without any assumption about the Concus-Finn Conjecture. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{remark}[theorem]{Remark} \newtheorem{claim}[theorem]{Claim} \section{Introduction} Consider the nonparametric prescribed mean curvature problem with contact angle boundary data in the cylinder $\Omega\times\mathbb{R}$ \begin{gather} \label{mean curvature} Nf = H(x,f) \quad\text{for } x\in \Omega \\ Tf \cdot \mathbf{\nu} = \cos \gamma \quad\text{on } \partial \Omega, \label{contactBCn} \end{gather} where $n\ge 2$, $\Omega\subset \mathbb{R}^{n}$ is bounded and open, $Tf = \nabla f/\sqrt{1 + |\nabla f|^{2}}$, $Nf = \nabla \cdot Tf$, $\mathbf{\nu}$ is the exterior unit normal on $\partial \Omega$, $\gamma: \partial\Omega\to [0,\pi]$ and $f\in C^{2}(\Omega)$. As in Part 1 of \cite{JeffresLan1A}, we consider variational solutions of (\ref{mean curvature})-(\ref{contactBCn}) and sequences $\{f_{j}\}$ which converge locally to generalized solutions $f_{\infty}:\Omega_{\infty}\to [-\infty,\infty]$ of (\ref{mean curvature})-(\ref{contactBCn}) of the functional \begin{equation} \label{JimXX} \mathcal{F}_{\infty} (g) = \int _{\Omega_{\infty}} \sqrt{1+ | Dg |^{2}} \,dx - \int_{\partial\Omega_{\infty}} \cos(\gamma_{\infty})g \,dH_{n} \end{equation} in the sense that for each compact subset $K$ of $\mathbb{R}^{n+1}$ with finite perimeter, $U_{\infty}$ minimizes the functional $F_{K}$ defined on subsets of $\Omega_{\infty}\times \mathbb{R}$ by $$ F_{K}(V)=\int_{K\cap (\Omega_{\infty}\times \mathbb{R})} |D\phi_{V}| - \int_{K\cap (\partial \Omega_{\infty}\times \mathbb{R})} \cos(\gamma_{\infty})\phi_{V}\,dH_{n}; $$ here $U_{\infty}=\{(x,t)\in \Omega_{\infty}\times \mathbb{R}:t2$ would be a difficult task which might have important applications to variational problems in $\mathbb{R}^{n}$; we hope our results serve as a first step in this direction. \section{Statement of Results} Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ with a corner at $O=(0,0)\in \partial \Omega$ such that, for some $\delta_{0}>0$, $\partial \Omega$ is piecewise smooth in $B_{\delta_{0}}(O)$ and $\partial \Omega \cap B_{\delta_{0}}(O)$ consists of two $C^{1, \lambda }$ arcs ${\partial}^{+}\Omega$ and $\partial^{-}\Omega,$ with \( \lambda \in (0,1), \) whose tangent lines approach the lines $L^{+}= \{\theta = \alpha\}$ and $L^{-}=\{ \theta = - \alpha\}$, respectively, as the point $O$ is approached. Let $\mathbf{\nu}^{+}$ and $\mathbf{\nu}^{-}$ denote the exterior unit normals on $\partial^{+} \Omega$ and $\partial^{-} \Omega$ respectively. Here we assume $\alpha\in (0,\pi)$, polar coordinates relative to $O$ are denoted by $r$ and $\theta$ and $B_{\delta}(O)$ is the ball in $\mathbb{R}^{2}$ of radius $\delta$ about $O$. Let $(x^{+}(s),y^{+}(s))$ be an arclength parametrization of ${\partial}^{+}\Omega$ and $(x^{-}(s),y^{-}(s))$ be an arclength parametrization of ${\partial}^{-}\Omega$, where $s=0$ corresponds to the point $O$ for both parametrizations. We will assume $\gamma_{1}=\lim_{s\downarrow 0}\gamma(x^{+}(s),y^{+}(s))$ and $\gamma_{2}=\lim_{s\downarrow 0}\gamma(x^{-}(s),y^{-}(s))$ both exist and $\gamma_{1}, \gamma_{2} \in (0,\pi)$. In this case, $$ \Omega_{\infty} =\{(r\cos \theta, r\sin \theta):r>0, -\alpha<\theta<\alpha\}, $$ $(\partial \Omega_{\infty})\setminus \{O\}=\Sigma_{1} \cup \Sigma_{2}$ with \begin{gather*} \Sigma_{j} =\{(r\cos \theta, r\sin \theta):r>0, \theta=(-1)^{j+1}\alpha\}, \quad j=1,2, \\ \lim_{s\downarrow 0} \nu^{+}(s)=\nu_{1}=(-\sin(\alpha),\cos(\alpha)), \quad \lim_{s\downarrow 0} \nu^{-}(s)=\nu_{2}=(-\sin(\alpha),-\cos(\alpha)) \end{gather*} and the limiting contact angle $\gamma_{\infty}$ equals $\gamma_{1}$ on $\Sigma_{1}$ and $\gamma_{2}$ on $\Sigma_{2}$. A set $\mathcal{P}\subset \Omega_{\infty}$ minimizes $\Phi$ if and only if for each $T>0$, $$ \Phi_{T} (\mathcal{P}) \le \Phi_{T} (\mathcal{P} \cup S) \quad \text{and} \quad \Phi_{T} (\mathcal{P}) \le \Phi_{T} (\mathcal{P} \setminus S) \quad \text{for every } S\subset \Omega_{\infty}^{T}\ , $$ where $\Omega_{\infty}^{T}= \overline{B_{T}(O)}\cap \Omega_{\infty}$, $\Sigma_{j}^{T}= \overline{B_{T}(O)}\cap \Sigma_{j}$, $j=1,2$, and \begin{align*} \Phi_{T} (A)&= \int _{\Omega_{\infty}^{T}} | D \phi_{A} | - \cos (\gamma_{1}) \int _{\Sigma_{1}^{T}} \phi_{A} d H^{1} - \cos (\gamma_{2}) \int _{\Sigma_{2}^{T}} \phi_{A} d H^{1}\\ &= H^{1} (\Omega_{\infty}^{T} \cap \partial A) - \cos (\gamma_{1}) H^{1} (\Sigma_{1}^{T} \cap \partial A) - \cos (\gamma_{2}) H^{1} (\Sigma_{2}^{T} \cap \partial A)\ . \end{align*} A set $\mathcal{N}\subset \Omega_{\infty}$ minimizes $\Psi$ if and only if for each $T>0$, $$ \Psi_{T} (\mathcal{N}) \le \Psi_{T} (\mathcal{N} \cup S) \quad \text{and} \quad \Psi_{T} (\mathcal{N}) \le \Psi_{T} (\mathcal{N} \setminus S) \quad\text{for every } S\subset \Omega_{\infty}^{T} $$ where \begin{align*} \Psi_{T} (A) &= \int _{\Omega_{\infty}^{T}} | D \phi_{A} | + \cos (\gamma_{1}) \int _{\Sigma_{1}^{T}} \phi_{A} d H^{1} + \cos (\gamma_{2}) \int _{\Sigma_{2}^{T}} \phi_{A} d H^{1}\\ &= H^{1} (\Omega_{\infty}^{T} \cap \partial A) + \cos (\gamma_{1}) H^{1} (\Sigma_{1}^{T} \cap \partial A) + \cos (\gamma_{2}) H^{1} (\Sigma_{2}^{T} \cap \partial A). \end{align*} If $\mathcal{P}$ minimizes $\Phi$, then after modification on a set of measure zero, we may assume $\partial \mathcal{P}$ coincides with the essential boundary of $\mathcal{P}$ (e.g. \cite[Theorem 1.1]{Gui:80}) and $\Omega_{\infty}\cap\partial \mathcal{P}$ consists of a union of rays. If $\mathcal{N}$ minimizes $\Psi$, then the same holds for $\partial \mathcal{N}$ and $\Omega_{\infty}\cap\partial \mathcal{N}$. We may also assume $\mathcal{P}$ and $\mathcal{N}$ are open. In the following theorems, we determine the geometric shapes of $\mathcal{P}$ (Theorem \ref{thm4}) and $\mathcal{N}$ (Theorem \ref{thm5}); cases (viii) and (xi) are special cases of (x) and (xiii) respectively and are included separately to assist in the descriptions of cases (ix) and (xii). To illustrate these geometries, we provide Figures \ref{Combo2} and \ref{Combo3}; cases (viii) and (xi) in Figure \ref{Combo2} are special cases of (x) and (xiii) respectively and are included separately to illustrate cases (ix) and (xii). The shaded regions in these figures illustrate $\mathcal{P}$ and the unshaded regions illustrate $\mathcal{N}$; we note that these figures should be interpreted independently and, while $\mathcal{P}$ and $\mathcal{N}$ must be disjoint, it is not true in general that $\overline{\mathcal{P}}\cup \overline{\mathcal{N}}\in \{\emptyset, \overline{\Omega_{\infty}}\}$. This is illustrated by Scherk or skewed Scherk surfaces. For example, let $a>0$ and set $$ f(x,y)=\frac{1}{a}\big( \ln(\sin(ax)) - \ln(\sin(ay)) \big) \quad \text{if } 0\frac{\pi}{a} \\ \pi &\text{if }x=0,0\frac{\pi}{a} \end{cases} $$ and $u:\Omega\to [-\infty,\infty]$ defined by $$ u(x,y) = \begin{cases} \infty &\text{if } (x,y)\in \Omega_{1} \\ f(x,y) &\text{if } (x,y)\in \Omega_{0} \\ -\infty &\text{if }(x,y)\in \Omega_{2}. \end{cases} $$ Notice that $u$ is a generalized solution of (\ref{mean curvature})-(\ref{contactBCn}) with $H\equiv 0$ and the sets $\mathcal{P}=\{(x,y):u(x,y)=\infty\}$ and $\mathcal{N}=\{(x,y):u(x,y)=-\infty\}$ are $\Omega_{1}$ and $\Omega_{2}$ respectively (recall that we require $\mathcal{P}$ and $\mathcal{N}$ to be open). We can, of course, modify this example so the domain $\Omega$ is convex. Set $\Omega=\{ (x,y) : 0\pi/2$ and $\mathcal{P}\subset\Omega_{\infty}$ minimizes $\Phi$. Let $(r,\theta)$ be polar coordinates about $O$. Then one of the following holds: \smallskip \noindent {\rm(i)} $\mathcal{P}=\emptyset$ or $\mathcal{P}=\Omega_{\infty}$; \smallskip \noindent {\rm(ii)} $\gamma_{1}-\gamma_{2} \le 2\alpha-\pi$, there exists $A\in \Sigma_{1}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P} =\Sigma_{1}\setminus OA$, $\Omega_{\infty} \cap \partial\mathcal{ P}$ is the ray $L$ in $\Omega_{\infty}$ starting at $A$ and making an angle of measure $\gamma_{1}$ with $\Sigma_{1}\setminus OA$ and $\mathcal{P}$ is the open sector between $\Sigma_{1}\setminus OA$ and $L$; \smallskip \noindent {\rm(iii)} $\gamma_{1}-\gamma_{2} \ge \pi-2\alpha$, there exists $A\in \Sigma_{1}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P} =\Sigma_{2}\cup \overline{OA}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is the ray $L$ in $\Omega_{\infty}$ starting at $A$ and making an angle of measure $\gamma_{1}$ with $OA$ and $\mathcal{P}$ is the open region whose boundary is $\Sigma_{2} \cup \overline{OA} \cup L$; \smallskip \noindent {\rm(iv)} $\gamma_{1}-\gamma_{2} \le 2\alpha-\pi$, there exists $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P} =\Sigma_{1}\cup \overline{OB}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is the ray $L$ in $\Omega_{\infty}$ starting at $B$ and making an angle of measure $\gamma_{2}$ with $OB$ and $\mathcal{P}$ is the open region whose boundary is $\Sigma_{1} \cup \overline{OA} \cup L$; \smallskip \noindent {\rm(v)} $\gamma_{1}-\gamma_{2} \ge \pi-2\alpha$, there exists $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P} =\Sigma_{2}\setminus OB$, $\Omega_{\infty} \cap \partial\mathcal{ P}$ is the ray $L$ in $\Omega_{\infty}$ starting at $B$ and making an angle of measure $\gamma_{2}$ with $\Sigma_{2}\setminus OB$ and $\mathcal{P}$ is the open sector between $\Sigma_{2}\setminus OB$ and $L$; \smallskip \noindent {\rm(vi)} $\gamma_{1}+\pi-\gamma_{2} \le 2\alpha$, $\partial\Omega_{\infty}\cap\partial\mathcal{P} =\Sigma_{1}\cup \{O\}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is a ray $L=\{\theta=\beta\}$ in $\Omega_{\infty}$ starting at $O$ which makes an angle of measure greater than or equal to $\gamma_{1}$ with $\Sigma_{1}$ and an angle of measure greater than or equal to $\pi-\gamma_{2}$ with $\Sigma_{2}$ (i.e. $\pi-\alpha-\gamma_{2}\le \beta \le \alpha-\gamma_{1})$ and $\mathcal{P}=\{\beta<\theta<\alpha\}$; \smallskip \noindent {\rm(vii)} $\gamma_{2}+\pi-\gamma_{1} \le 2\alpha$, $\partial\Omega_{\infty}\cap\partial\mathcal{P} =\Sigma_{2}\cup \{O\}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is a ray $L=\{\theta=\beta\}$ in $\Omega_{\infty}$ starting at $O$ which makes an angle of measure greater than or equal to $\pi-\gamma_{1}$ with $\Sigma_{1}$ and an angle of measure greater than or equal to $\gamma_{2}$ with $\Sigma_{2}$ (i.e. $\gamma_{2}-\alpha \le \beta \le \alpha+\gamma_{1}-\pi)$ and $\mathcal{P}=\{-\alpha<\theta<\beta\}$; \smallskip \noindent {\rm(viii)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, $\partial\mathcal{P}$ is a line $L=\{\theta=\beta\}\cup \{\theta=\beta+\pi\}$ which passes through $O$ and makes angles of measure greater than or equal to $\pi-\gamma_{1}$ with $\Sigma_{1}$ and $\pi-\gamma_{2}$ with $\Sigma_{2}$ (i.e. $\pi-\alpha-\gamma_{2} \le \beta \le \alpha+\gamma_{1}-2\pi$) and $\mathcal{ P}=\{\beta<\theta<\beta+\pi \}$ is the component of $\Omega_{\infty}\setminus L$ whose closure is disjoint from $\Sigma_{1}\cup\Sigma_{2}$; \smallskip \noindent {\rm(ix)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, $\partial\mathcal{P}$ is a line $M$ in $\Omega_{\infty}$ which is a parallel translate of the line $L$ described in (viii) and $\mathcal{P}$ is the component of $\Omega_{\infty}\setminus M$ whose closure is disjoint from $\Sigma_{1}\cup\Sigma_{2}$; \smallskip \noindent {\rm(x)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, $\partial\Omega_{\infty}\cap\partial\mathcal{P} = \{O\}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is a pair of rays $L=\{\theta=\beta_{1}\}$ and $M=\{\theta=\beta_{2}\}$ in $\Omega_{\infty}$, each starting at $O$, such that $\beta_{1}-\beta_{2}\ge \pi$, $\alpha-\beta_{1}\ge \pi-\gamma_{1}$, $\beta_{2}+\pi\ge \pi-\gamma_{2}$, and $\mathcal{P}=\{\beta_{2}<\theta<\beta_{1}\}$; \smallskip \noindent {\rm(xi)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, $\partial\mathcal{P}$ is a line $L=\{\theta=\beta\}\cup \{\theta=\beta+\pi\}$ which passes through $O$ and makes angles of measure greater than or equal to $\gamma_{1}$ with $\Sigma_{1}$ and $\gamma_{2}$ with $\Sigma_{2}$ and $\mathcal{ P}=\{-\alpha<\theta<\beta \} \cup \{\beta+\pi < \theta < \alpha\}$ is the union of the (two) components of $\Omega_{\infty}\setminus L$ whose closures intersect $\Sigma_{1}\cup\Sigma_{2}$; \smallskip \noindent {\rm(xii)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, $\partial\mathcal{P}$ is a line $M$ in $\Omega_{\infty}$ which is a parallel translate of the line $L$ described in (xi) and $\mathcal{P}$ is the component of $\Omega_{\infty}\setminus M$ whose closure contains $\Sigma_{1}\cup\Sigma_{2}$; \smallskip \noindent {\rm(xiii)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, $\partial\Omega_{\infty}\cap\partial\mathcal{P} = \partial\Omega_{\infty}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is a pair of rays $L=\{\theta=\beta_{1}\}$ and $M=\{\theta=\beta_{2}\}$ in $\Omega_{\infty}$, each starting at $O$, such that $\beta_{1}-\beta_{2}\ge \pi$, $\alpha-\beta_{1}\ge \gamma_{1}$, $\beta_{2}+\pi\ge \gamma_{2}$, and $\mathcal{ P}=\mathcal{P}_{1} \cup \mathcal{P}_{2}$, where $\mathcal{ P}_{1}=\{\beta_{1}<\theta<\alpha\}$ and $\mathcal{ P}_{2}=\{-\alpha<\theta<\beta_{2}\}$; \smallskip \noindent {\rm(xiv)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, there exist $A\in \Sigma_{1}$ and $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P}= (\Sigma_{1}\setminus OA) \cup (\Sigma_{2}\setminus OB)$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is the union of rays $L_{1}$ and $L_{2}$ in $\Omega_{\infty}$, where $L_{1}$ starts at $A$ and makes an angle of measure $\gamma_{1}$ with $\Sigma_{1}\setminus OA$ and $L_{2}$ starts at $B$ and makes an angle of measure $\gamma_{2}$ with $\Sigma_{2}\setminus OB$, and $\mathcal{P}$ is the union of the open sectors between $\Sigma_{1}\setminus OA$ and $L_{1}$ and between $\Sigma_{2}\setminus OB$ and $L_{2}$ respectively; or \smallskip \noindent {\rm(xv)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, there exist $A\in \Sigma_{1}$ and $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P}= \overline{OA} \cup \overline{OB}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is the union of rays $L_{1}$ and $L_{2}$ in $\Omega_{\infty}$, where $L_{1}$ starts at $A$ and makes an angle of measure $\gamma_{1}$ with $OA$ and $L_{2}$ starts at $B$ and makes an angle of measure $\gamma_{2}$ with $OB$, and $\mathcal{P}$ is the open region in $\Omega_{\infty}$ between $L_{1}$ and $L_{2}$. \smallskip \noindent {\rm(xvi)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, there exists $A\in \Sigma_{1}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P}=\partial\Omega_{\infty}\setminus OA$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is a pair of rays $L$ and $M$ in $\Omega_{\infty}$, and $\mathcal{P}=\mathcal{ P}_{1} \cup \mathcal{P}_{2}$, where $L$ starts at $A$ and makes an angle of measure $\gamma_{1}$ with $\Sigma_{1}\setminus OA$, $M=\{\theta=\beta_{2}\}$ starts at $O$ with $-\alpha+\gamma_{2}\le \beta_{2}\le \alpha-\gamma_{1}-\pi$, $\mathcal{P}_{1}$ is the open, connected region in $\Omega_{\infty}$ with boundary $L\cup\Sigma_{1}\setminus OA$ and $\mathcal{P}_{2}=\{-\alpha<\theta<\beta_{2}\}$; \smallskip \noindent {\rm(xvii)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, there exists $A\in \Sigma_{1}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P}=\overline{OA}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is a pair of rays $L$ and $M$ in $\Omega_{\infty}$ and $\mathcal{P}$ is the connected open subset of $\Omega_{\infty}$ with boundary $L\cup \overline{OA}\cup M$, where $L$ starts at $A$ and makes an angle of measure $\gamma_{1}$ with $OA$ and $M=\{\theta=\beta_{2}\}$ starts at $O$ with $-\alpha+\pi-\gamma_{2}\le \beta_{2}\le \alpha+\gamma_{1}-2\pi$; \smallskip \noindent {\rm(xviii)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, there exists $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P} =\partial\Omega_{\infty}\setminus OB$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is a pair of rays $L$ and $M$ in $\Omega_{\infty}$ and $\mathcal{P}=\mathcal{P}_{1} \cup \mathcal{P}_{2}$, where $L=\{\theta=\beta_{1}\}$ starts at $O$ with $-\alpha+\gamma_{2}+\pi\le \beta_{1}\le \alpha-\gamma_{1}$, $M$ starts at $B$ and makes an angle of measure $\gamma_{2}$ with $\Sigma_{2}\setminus OB$, $\mathcal{ P}_{1}=\{\beta_{1}<\theta<\alpha\}$ and $\mathcal{P}_{2}$ is the open, connected region in $\Omega_{\infty}$ with boundary $L\cup\Sigma_{2}\setminus OB$; \smallskip \noindent {\rm(xix)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, there exists $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{P}=\overline{OB}$, $\Omega_{\infty} \cap \partial\mathcal{P}$ is a pair of rays $L$ and $M$ in $\Omega_{\infty}$, and $\mathcal{P}$ is the connected open subset of $\Omega_{\infty}$ with boundary $L\cup \overline{OB}\cup M$, where $L=\{\theta=\beta_{1}\}$ starts at $O$ with $-\alpha+2\pi-\gamma_{2}\le \beta_{1}\le \alpha+\gamma_{1}-\pi$ and $M$ starts at $B$ and makes an angle of measure $\gamma_{2}$ with $OB$. \end{theorem} \begin{theorem} \label{thm5} Suppose $\alpha>\pi/2$ and $\mathcal{N}\subset \Omega_{\infty}$ minimizes $\Psi$. Let $(r,\theta)$ be polar coordinates about $O$. Then one of the following holds: \smallskip \noindent {\rm(i)} $\mathcal{N}=\emptyset$ or $\mathcal{ N}=\Omega_{\infty}$; \smallskip \noindent {\rm(ii)} $\gamma_{1}-\gamma_{2} \le 2\alpha-\pi$, there exists $A\in \Sigma_{1}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N} =\Sigma_{2}\cup \overline{OA}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is the ray $L$ in $\Omega_{\infty}$ starting at $A$ and making an angle of measure $\pi-\gamma_{1}$ with $OA$ and $\mathcal{N}$ is the open region whose boundary is $\Sigma_{2} \cup \overline{OA} \cup L$; \smallskip \noindent {\rm(iii)} $\gamma_{1}-\gamma_{2} \ge \pi-2\alpha$, there exists $A\in \Sigma_{1}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N} =\Sigma_{1}\setminus OA$, $\Omega_{\infty} \cap \partial\mathcal{ N}$ is the ray $L$ in $\Omega_{\infty}$ starting at $A$ and making an angle of measure $\pi-\gamma_{1}$ with $\Sigma_{1}\setminus OA$ and $\mathcal{N}$ is the open sector between $\Sigma_{1}\setminus OA$ and $L$; \smallskip \noindent {\rm(iv)} $\gamma_{1}-\gamma_{2} \le 2\alpha-\pi$, there exists $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N} =\Sigma_{2}\setminus OB$, $\Omega_{\infty} \cap \partial\mathcal{ N}$ is the ray $L$ in $\Omega_{\infty}$ starting at $B$ and making an angle of measure $\pi-\gamma_{2}$ with $\Sigma_{2}\setminus OB$ and $\mathcal{N}$ is the open sector between $\Sigma_{2}\setminus OB$ and $L$; \smallskip \noindent {\rm(v)} $\gamma_{1}-\gamma_{2} \ge \pi-2\alpha$, there exists $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N} =\Sigma_{1}\cup \overline{OB}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is the ray $L$ in $\Omega_{\infty}$ starting at $B$ and making an angle of measure $\pi-\gamma_{2}$ with $OB$ and $\mathcal{N}$ is the open region whose boundary is $\Sigma_{1} \cup \overline{OA} \cup L$; \smallskip \noindent {\rm(vi)} $\gamma_{1}+\pi-\gamma_{2} \le 2\alpha$, $\partial\Omega_{\infty}\cap\partial\mathcal{N} =\Sigma_{2}\cup \{O\}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is a ray $L=\{\theta=\beta\}$ in $\Omega_{\infty}$ starting at $O$ which makes an angle of measure greater than or equal to $\gamma_{1}$ with $\Sigma_{1}$ and an angle of measure greater than or equal to $\pi- \gamma_{2}$ with $\Sigma_{2}$ (i.e. $\pi-\alpha-\gamma_{2}\le \beta \le \alpha-\gamma_{1})$ and $\mathcal{N}=\{-\alpha<\theta<\beta\}$; \smallskip \noindent {\rm(vii)} $\gamma_{2}+\pi-\gamma_{1} \le 2\alpha$, $\partial\Omega_{\infty}\cap\partial\mathcal{N} =\Sigma_{1}\cup \{O\}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is a ray $L=\{\theta=\beta\}$ in $\Omega_{\infty}$ starting at $O$ which makes an angle of measure greater than or equal to $\pi-\gamma_{1}$ with $\Sigma_{1}$ and an angle of measure greater than or equal to $\gamma_{2}$ with $\Sigma_{2}$ (i.e. $\gamma_{2}-\alpha \le \beta \le \alpha+\gamma_{1}-\pi)$ and $\mathcal{N}=\{\beta<\theta<\alpha\}$; \smallskip \noindent {\rm(viii)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, $\partial\mathcal{N}$ is a line $L=\{\theta=\beta\}\cup \{\theta=\beta+\pi\}$ which passes through $O$ and makes angles of measure greater than or equal to $\pi-\gamma_{1}$ with $\Sigma_{1}$ and $\pi-\gamma_{2}$ with $\Sigma_{2}$ (i.e. $\pi-\alpha-\gamma_{2}\le \beta \le \alpha+\gamma_{1}-2\pi)$ and $\mathcal{N}=\{-\alpha<\theta<\beta \} \cup \{\beta+\pi < \theta < \alpha\}$ is the union of the (two) components of $\Omega_{\infty}\setminus L$ whose closures intersect $\Sigma_{1}\cup\Sigma_{2}$; \smallskip \noindent {\rm(ix)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, $\partial\mathcal{N}$ is a line $M$ in $\Omega_{\infty}$ which is a parallel translate of the line $L$ described in (viii) and $\mathcal{N}$ is the component of $\Omega_{\infty}\setminus M$ whose closure contains $\Sigma_{1}\cup\Sigma_{2}$; \smallskip \noindent {\rm(x)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, $\partial\Omega_{\infty}\cap\partial\mathcal{N} = \partial\Omega_{\infty}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is a pair of rays $L=\{\theta=\beta_{1}\}$ and $M=\{\theta=\beta_{2}\}$ in $\Omega_{\infty}$, each starting at $O$, such that $\beta_{1}-\beta_{2}\ge \pi$, $\alpha-\beta_{1}\ge \pi-\gamma_{1}$, $\beta_{2}+\pi\ge \pi-\gamma_{2}$, and $\mathcal{N}=\mathcal{N}_{1} \cup \mathcal{N}_{2}$, where $\mathcal{N}_{1}=\{\beta_{1}<\theta<\alpha\}$ and $\mathcal{ N}_{2}=\{-\alpha<\theta<\beta_{2}\}$; \smallskip \noindent {\rm(xi)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, $\partial\mathcal{N}$ is a line $L=\{\theta=\beta\}\cup \{\theta=\beta+\pi\}$ which passes through $O$ and makes angles of measure greater than or equal to $\gamma_{1}$ with $\Sigma_{1}$ and $\gamma_{2}$ with $\Sigma_{2}$ (i.e. $\gamma_{2}-\alpha \le \beta \le \alpha-\gamma_{1}-\pi$) and $\mathcal{N}=\{\beta<\theta<\beta+\pi \}$ is the component of $\Omega_{\infty}\setminus L$ whose closure is disjoint from $\Sigma_{1}\cup\Sigma_{2}$; \smallskip \noindent {\rm(xii)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, $\partial\mathcal{N}$ is a line $M$ in $\Omega_{\infty}$ which is a parallel translate of the line $L$ described in (xi) and $\mathcal{N}$ is the component of $\Omega_{\infty}\setminus M$ whose closure is disjoint from $\Sigma_{1}\cup\Sigma_{2}$; \smallskip \noindent {\rm(xiii)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, $\partial\Omega_{\infty}\cap\partial\mathcal{N} = \{O\}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is a pair of rays $L=\{\theta=\beta_{1}\}$ and $M=\{\theta=\beta_{2}\}$ in $\Omega_{\infty}$, each starting at $O$, such that $\beta_{1}-\beta_{2}\ge \pi$, $\alpha-\beta_{1}\ge \gamma_{1}$, $\beta_{2}+\alpha\ge \gamma_{2}$, and $\mathcal{ N}=\{\beta_{2}<\theta<\beta_{1}\}$; \smallskip \noindent {\rm(xiv)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, there exist $A\in \Sigma_{1}$ and $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N}= \overline{OA} \cup \overline{OB}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is the union of rays $L_{1}$ and $L_{2}$ in $\Omega_{\infty}$, where $L_{1}$ starts at $A$ and makes an angle of measure $\gamma_{1}$ with $\Sigma_{1}\setminus OA$ and $L_{2}$ starts at $B$ and makes an angle of measure $\gamma_{2}$ with $\Sigma_{2}\setminus OB$, and $\mathcal{N}$ is the open region in $\Omega_{\infty}$ between $L_{1}$ and $L_{2}$; or \smallskip \noindent {\rm(xv)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, there exist $A\in \Sigma_{1}$ and $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N}= (\Sigma_{1}\setminus OA) \cup (\Sigma_{2}\setminus OB)$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is the union of rays $L_{1}$ and $L_{2}$ in $\Omega_{\infty}$, where $L_{1}$ starts at $A$ and makes an angle of measure $\gamma_{1}$ with $OA$ and $L_{2}$ starts at $B$ and makes an angle of measure $\gamma_{2}$ with $OB$, and $\mathcal{N}$ is the union of the open sectors between $\Sigma_{1}\setminus OA$ and $L_{1}$ and between $\Sigma_{2}\setminus OB$ and $L_{2}$ respectively. \smallskip \noindent {\rm(xvi)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, there exists $A\in \Sigma_{1}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N}=\partial\Omega_{\infty}\setminus OA$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is a pair of rays $L$ and $M$ in $\Omega_{\infty}$, and $\mathcal{N}$ is the connected open subset of $\Omega_{\infty}$ with boundary $L\cup \overline{OA}\cup M$, where $L$ starts at $A$ and makes an angle of measure $\gamma_{1}$ with $\Sigma_{1}\setminus OA$ and $M=\{\theta=\beta_{2}\}$ starts at $O$ with $-\alpha+\gamma_{2}\le \beta_{2}\le \alpha-\gamma_{1}-\pi$; \smallskip \noindent {\rm(xvii)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, there exists $A\in \Sigma_{1}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N}=\overline{OA}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is a pair of rays $L$ and $M$ in $\Omega_{\infty}$ and $\mathcal{N}=\mathcal{N}_{1} \cup \mathcal{N}_{2}$, where $L$ starts at $A$ and makes an angle of measure $\gamma_{1}$ with $OA$, $M=\{\theta=\beta_{2}\}$ starts at $O$ with $-\alpha+\pi-\gamma_{2}\le \beta_{2}\le \alpha+\gamma_{1}-2\pi$, $\mathcal{N}_{1}$ is the connected open subset of $\Omega_{\infty}$ with boundary $L\cup \Sigma_{2}\setminus {OA}$ and $\mathcal{ N}_{2}=\{-\alpha<\theta<\beta_{2}\}$; \smallskip \noindent {\rm(xviii)} $\gamma_{1} + \gamma_{2}\le 2\alpha-\pi$, there exists $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N} =\partial\Omega_{\infty}\setminus OB$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is a pair of rays $L$ and $M$ in $\Omega_{\infty}$ and $\mathcal{N}$ is the connected open subset of $\Omega_{\infty}$ with boundary $L\cup \overline{OB}\cup M$, where $L=\{\theta=\beta_{1}\}$ starts at $O$ with $-\alpha+\gamma_{2}+\pi\le \beta_{1}\le \alpha-\gamma_{1}$ and $M$ starts at $B$ and makes an angle of measure $\gamma_{2}$ with $\Sigma_{2}\setminus OB$; \smallskip \noindent {\rm(xix)} $\pi-\gamma_{1} + \pi-\gamma_{2}\le 2\alpha-\pi$, there exists $B\in \Sigma_{2}$ such that $\partial\Omega_{\infty}\cap\partial\mathcal{N}=\overline{OB}$, $\Omega_{\infty} \cap \partial\mathcal{N}$ is a pair of rays $L$ and $M$ in $\Omega_{\infty}$, and $\mathcal{N}=\mathcal{N}_{1} \cup \mathcal{N}_{2}$, where $L=\{\theta=\beta_{1}\}$ starts at $O$ with $-\alpha+2\pi-\gamma_{2}\le \beta_{1}\le \alpha+\gamma_{1}-\pi$, $M$ starts at $B$ and makes an angle of measure $\gamma_{2}$ with $OB$, $\mathcal{ N}_{1}=\{\beta_{2}<\theta<\alpha\}$ and $\mathcal{N}_{2}$ is the connected open subset of $\Omega_{\infty}$ with boundary $M\cup \Sigma_{2}\setminus {OB}$. \end{theorem} \begin{figure}[ht!] \begin{center} \begin{picture}(0,0) \includegraphics{fig1} %Combo2.pstex_t \end{picture} \setlength{\unitlength}{2368sp} \begin{picture}(8424,14574)(1189,-14923) \put(1826,-1861){$A$} \put(2350,-1936){$O$} \put(4826,-1861){$A$} \put(5350,-1936){$O$} \put(7820,-2011){$B$} \put(8350,-1936){$O$} \put(1750,-3736){case (ii)} \put(4600,-3736){case (iii)} \put(7500,-3736){case (iv)} \put(2420,-5536){$O$} \put(1826,-5620){$B$} \put(5400,-5536){$O$} \put(8420,-5536){$O$} \put(1750,-7411){case (v)} \put(4620,-7411){case (vi)} \put(7551,-7411){case (vii)} \put(8400,-9136){$O$} \put(5400,-9136){$O$} \put(2400,-9136){$O$} \put(1700,-11086){case (viii)} \put(4601,-11086){case (ix)} \put(7526,-11086){case (x)} \put(2400,-12736){$O$} \put(5400,-12736){$O$} \put(8400,-12736){$O$} \put(1900,-14836){case (xi)} \put(4750,-14761){case (xii)} \put(7900,-14761){case (xiii)} \end{picture} \end{center} \caption{Theorems \ref{thm4} and \ref{thm5}} \label{Combo2} \end{figure} \begin{figure}[ht] \begin{center} \begin{picture}(0,0) \includegraphics{fig2} % Combo3.pstex \end{picture} \setlength{\unitlength}{2368sp} \begin{picture}(8424,7449)(1189,-7798) \put(1750,-1800){$A$} \put(1750,-2100){$B$} \put(2420,-1936){$O$} \put(1800,-3736){case (xiv)} \put(4726,-1800){$A$} \put(4726,-2100){$B$} \put(5400,-1936){$O$} \put(4620,-3736){case (xv)} \put(7650,-1786){$A$} \put(8450,-1936){$O$} \put(7651,-3736){case (xvi)} \put(1576,-5686){$A$} \put(2400,-5911){$O$} \put(1726,-7636){case (xvii)} \put(5026,-6211){$B$} \put(5320,-5911){$O$} \put(4601,-7636){case (xviii)} \put(7950,-6211){$B$} \put(8370,-5911){$O$} \put(7726,-7561){case (xix)} \end{picture} \end{center} \caption{Theorems \ref{thm4} and \ref{thm5}} \label{Combo3} \end{figure} \begin{corollary} \label{coro1} Suppose $\alpha>\pi/2$ and $(\gamma_{1},\gamma_{2})$ satisfies $\gamma_{1}-\gamma_{2}<\pi-2\alpha$ (i.e. $(\gamma_{1},\gamma_{2})$ lies in the open region denoted $D^{+}_{2}$ in Figure \ref{NC_CFR}). Then only cases (i), (ii), (iv) and (vi) of Theorems \ref{thm4} and \ref{thm5} can hold. \end{corollary} \begin{corollary} \label{coro2} Suppose $\alpha>\pi/2$ and $(\gamma_{1},\gamma_{2})$ satisfies $\gamma_{1}-\gamma_{2}>2\alpha-\pi$ (i.e. $(\gamma_{1},\gamma_{2})$ lies in the open region denoted $D^{-}_{2}$ in Figure \ref{NC_CFR}). Then only cases (i), (iii), (v) and (vii) of Theorems \ref{thm4} and \ref{thm5} can hold. \end{corollary} \begin{corollary} \label{coro3} Suppose $\alpha>\pi/2$ and $(\gamma_{1},\gamma_{2})$ satisfies $\gamma_{1}+\gamma_{2}<2\alpha-\pi$ (i.e. $(\gamma_{1},\gamma_{2})$ lies in the open region denoted $D^{+}_{1}$ in Figure \ref{NC_CFR}). Then cases (viii), (ix), (x), (xv), (xvii) and (xix) of Theorems \ref{thm4} and \ref{thm5} cannot hold. \end{corollary} \begin{corollary} \label{coro4} Suppose $\alpha>\pi/2$ and $(\gamma_{1},\gamma_{2})$ satisfies $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$ (i.e. $(\gamma_{1},\gamma_{2})$ lies in the open region denoted $D^{-}_{1}$ in Figure \ref{NC_CFR}). Then cases (xi), (xii), (xiii), (xiv), (xvi) and (xviii) of Theorems \ref{thm4} and \ref{thm5} cannot hold. \end{corollary} The proofs of these corollaries are simple exercises in checking angles. \section{Applications to capillarity} Consider the stationary liquid-gas interface formed by an incompressible fluid in a vertical cylindrical tube with cross-section $\Omega$. For simplicity, we assume that near $(0,0)$, $\partial\Omega$ has straight sides (as in \cite{Shi2}) and so we may assume \begin{equation} \label{yazoo} \Omega=\{(r\cos(\theta), r\sin(\theta)) : 0 3\pi-2\alpha\}$ \noindent $\mathcal{D}_{2}^{+} = \{ (\gamma_{1},\gamma_{2}): \gamma_{1}-\gamma_{2}<\pi-2\alpha \}$ \noindent $\mathcal{D}_{2}^{-} = \{ (\gamma_{1},\gamma_{2}): \gamma_{1}-\gamma_{2} > 2\alpha-\pi \}$. \begin{figure}[ht] \begin{center} \begin{picture}(0,0) \includegraphics{fig3} %CFrect2.pstex \end{picture} \setlength{\unitlength}{1973sp} \begin{picture}(8730,8334)(1036,-9343) \put(1680,-1786){$\gamma_2$} \put(9301,-8986){$\gamma_1$} \put(5626,-5236){$\mathcal{R}$} \put(9751,-6511){$2\pi-2\alpha$} \put(4126,-1336){$2\pi-2\alpha$} \put(6601,-9211){$2\alpha-\pi$} \put(1051,-4036){$2\alpha-\pi$} \put(2851,-2536){$\mathcal{D}^{+}_2$} \put(7726,-3511){$\mathcal{D}^{-}_1$} \put(3301,-7261){$\mathcal{D}^{+}_1$} \put(8401,-8161){$\mathcal{D}^{-}_2$} \end{picture} \end{center} \caption{Nonconvex Concus-Finn rectangle} \label{NC_CFR} \end{figure} Shi assumed the Concus-Finn Conjecture was true for $\gamma_{1}\in \{0,\pi \}$ and $\gamma_{2}\in \{0,\pi \})$ and proved in \cite{Shi} and \cite{Shi2} that a solution $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ of \eqref{capillary equation} and \eqref{contactBC} must be discontinuous at $O$ when $(\gamma_{1},\gamma_{2})\in \mathcal{ D}_{1}^{+} \cup \mathcal{D}_{1}^{-} \cup \mathcal{D}_{2}^{+} \cup \mathcal{D}_{2}^{-}$ and $\kappa>0$. Our goal here is to reach the same conclusion when $\kappa=\lambda=0$ and to prepare the necessary background for a direct proof of the ``nonconvex Concus-Finn conjecture'' in \cite{JL3}. To determine the behavior of $f$ near $(0,0)$, we need first to determine the behavior of the Gauss map on the edge $\{ (0,0,z):z\in \mathbb{R} \}$. For $\beta\in (-\alpha,\alpha)$, let $t_{\beta}$ denote the set of sequences $(X_{j})$ in $\Omega$ which satisfy \begin{equation} \label{krogoth} \lim_{j\to\infty} X_{j}=(0,0) \quad \text{and} \quad \lim_{j\to\infty}\frac{X_{j}}{|X_{j}|} = (\cos(\beta),\sin(\beta)). \end{equation} For a given solution $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ of \eqref{capillary equation} and \eqref{contactBC}, we define \begin{equation} \label{rabbit} \vec n(x,y)=\vec n_{f}(x,y)=\big(Tf(x,y), \frac{-1}{\sqrt{1+|\nabla f(x,y)|^2}}\big) \end{equation} to be the (downward) unit normal to the graph of $f$ at $(x,y,f(x,y))$. Let $S_{0}^{2} = \{ (x,y,0):x,y\in \mathbb{R},\; x^2+y^2=1 \}$. \begin{lemma} \label{lem1} Suppose $\alpha > \pi/2$, $(\gamma_{1},\gamma_{2})\in D^{+}_{1}\cup D^{-}_{1} \cup D^{+}_{2}\cup D^{-}_{2}$, $f$ is a solution of \eqref{capillary equation} and \eqref{contactBC}, $\beta\in (-\alpha,\alpha)$, and $(X_{j}) \in t_{\beta}$ such that $\eta=\lim_{j\to\infty}\vec n_{f}(X_{j})$ exists. Then $\eta\in S_{0}^{2}$. \end{lemma} It is a fact that no nonvertical plane in $\mathbb{R}^3$ meets $L^{+}\times \mathbb{R}$ in an angle of $\gamma_{1}$ and $L^{-}\times \mathbb{R}$ in an angle of $\gamma_{2}$ when $(\gamma_{1},\gamma_{2})\in D^{+}_{1}\cup D^{-}_{1}\cup D^{+}_{2}\cup D^{-}_{2}$ in Figure \ref{NC_CFR}. The proof of the lemma follows as in the proof of \cite[Lemma 3.1]{JeffresLan1A}. \begin{remark} \label{rmk1} \rm As noted in \cite[Remark 3.2]{JeffresLan1A}, we may assume in this section that $\Omega$ and $\gamma$ are as described in \S 2 and $f$ satisfies \eqref{capillary equation} and \eqref{contactBC}. \end{remark} \begin{lemma} \label{lem2} Suppose $\alpha > \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{+}_{2}$ (i.e. $\gamma_{1}-\gamma_{2}<\pi-2\alpha$). Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC}. Let $\beta\in (-\alpha,\alpha)$ and let $\{ (x_{j},y_{j})\}\in t_{\beta}$. \begin{itemize} \item[(i)] If $\beta\in [-\alpha+\pi-\gamma_{2},\alpha-\gamma_{1}]$, then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = (-\sin(\beta), \cos(\beta),0)$. \item[(ii)] If $\beta\in (-\alpha,-\alpha+\pi-\gamma_{2}]$, then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = (-\sin(-\alpha+\pi-\gamma_{2}), \cos(-\alpha+\pi-\gamma_{2}),0)$. \item[(iii)] If $\beta\in [\alpha-\gamma_{1},\alpha)$, $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = (-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$. \end{itemize} \end{lemma} In light of Corollary \ref{coro1}, the proof of this lemma is essentially the same as that of \cite[Lemma 3.1]{JeffresLan1A}. \begin{lemma} \label{lem3} Suppose $\alpha > \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{-}_{2}$ (i.e. $\gamma_{1}-\gamma_{2}>2\alpha-\pi$). Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC}. Let $\beta\in (-\alpha,\alpha)$ and let $\{ (x_{j},y_{j})\}\in t_{\beta}$. \begin{itemize} \item[(i)] If $\beta\in [-\alpha+\gamma_{2},\alpha+\gamma_{1}-\pi]$, then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = (\sin(\beta), -\cos(\beta),0)$. \item[(ii)] If $\beta\in (-\alpha,-\alpha+\gamma_{2}]$, then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = (\sin(-\alpha+\gamma_{2}), -\cos(-\alpha+\gamma_{2}),0)$. \item[(iii)] If $\beta\in [\alpha+\gamma_{1}-\pi,\alpha)$, then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = (\sin(\alpha+\gamma_{1}-\pi), -\cos(\alpha+\gamma_{1}-\pi),0)$. \end{itemize} \end{lemma} In light of Corollary \ref{coro2}, the proof of this lemma follows using the techniques in the proof of \cite[Lemma 3.1]{JeffresLan1A}. \begin{lemma} \label{lem4} Suppose $\alpha > \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{+}_{1}\cup D^{-}_{1}$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC}. Then one of the following conclusions holds: \begin{itemize} \item[(i)] For each $\beta\in (-\alpha,-\alpha+\min\{\gamma_{2},\pi-\gamma_{2}\})$ and each sequence $(X_{j})\in t_{\beta}$, $$ \lim_{j\to \infty} \vec n_{f}(x_{j},y_{j}) = (\cos(\theta_{1}), \sin(\theta_{1}),0), $$ where $\theta_{1}= -\alpha - \gamma_{2}- \pi/2$. \item[(ii)] For each $\beta\in (-\alpha,-\alpha+\min\{\gamma_{2},\pi-\gamma_{2}\})$ and each sequence $(X_{j})\in t_{\beta}$, $$ \lim_{j\to \infty} \vec n_{f}(x_{j},y_{j}) = (\cos(\theta_{2}), \sin(\theta_{2}),0), $$ where $\theta_{2}= -\alpha + \gamma_{2}- \pi/2$. \end{itemize} \end{lemma} \begin{proof} We have $\gamma_{1}+\gamma_{2}\in (0,2\alpha-\pi)\cup (3\pi-2\alpha,2\pi)$. Let us define $$ C_{\beta}(f)=\{ \eta\in S^{2} :\eta=\lim_{j\to\infty} \vec n_{f}(X_{j}) \text{ for some } (X_{j})\in t_{\beta} \} $$ for each $\beta\in (-\alpha,\alpha)$ and set $C(f)=\cup_{\beta\in (-\alpha,\alpha)} C_{\beta}(f)$. From items (a) of Lemmas \ref{lem6}-\ref{lem13} (see \S 6) and a simple computation, we have $$ C_{\beta}(f) \subset \{ (\cos(\theta_{1}), \sin(\theta_{1}),0), \; (\cos(\theta_{2}), \sin(\theta_{2}),0) \}, $$ when $\beta\in (-\alpha,-\alpha+\min\{\gamma_{2},\pi-\gamma_{2}\} )$. We argue by contradiction and therefore assume there exist $\beta_{1},\beta_{2}\in (-\alpha,-\alpha+\min\{\gamma_{2},\pi-\gamma_{2}\} )$, $(X_{j})\in t_{\beta_{1}}$ and $(Y_{j})\in t_{\beta_{2}}$ such that \begin{gather*} \lim_{j\to\infty} \vec n_{f}(X_{j}) = (\cos(\theta_{1}), \sin(\theta_{1}),0), \\ \lim_{j\to\infty} \vec n_{f}(Y_{j}) = (\cos(\theta_{2}), \sin(\theta_{2}),0). \end{gather*} For each $j\in \mathbb{N}$, let $\sigma_{j}$ be the line segment joining $X_{j}$ and $Y_{j}$. By the Intermediate Value Theorem, there exists $Z_{j}\in \sigma_{j}$ such that \begin{equation} \label{bad beer} \vec n_{f}(Z_{j})\in \{ (r\cos(-\alpha-\frac{\pi}{2}), r\sin(-\alpha-\frac{\pi}{2}),-\sqrt{1-r^2}): -1\le r\le 1\} \end{equation} for each $j\in \mathbb{N}$. Since $\lim_{j\to\infty} X_{j}=(0,0)$ and $\lim_{j\to\infty} Y_{j}=(0,0)$, we see that $\lim_{j\to\infty} Z_{j}=(0,0)$. Using compactness and the argument in the proof of Lemma \ref{lem6} (see \S 6), we may replace the sequence $(Z_{j})$ by a subsequence such that $$ \lim_{j\to\infty}\frac{Z_{j}}{|Z_{j}|} = (\cos(\beta_{3}),\sin(\beta_{3})) \quad \text{for some } \beta_{3}\in (-\alpha, -\alpha +\min\{\gamma_{2},\pi-\gamma_{2}\}) $$ and $\lim_{j\to \infty} \vec n_{j}(Z_{j})=\eta$ exists and $\eta \in C_{\beta_{3}}(f)$, where $$ f_{j}(X)=\frac{f(|Z_{j}|X)-f(Z_{j})} {|Z_{j}|} $$ and $\vec n_{j}(x,y)$ is given by (\ref{free beer}). Now (\ref{bad beer}) and Lemma \ref{lem1} imply $$ \eta=\pm (\cos(-\alpha-\frac{\pi}{2}),\sin(-\alpha-\frac{\pi}{2}),0). $$ However, neither of these unit vectors lies in $C_{\beta_{3}}(f)$ and so a contradiction exists. Thus our claim is established. \end{proof} \begin{lemma} \label{lem5} Suppose $\alpha > \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{+}_{1}\cup D^{-}_{1}$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC}. Then one of the following conclusions holds: \begin{itemize} \item[(i)] For each $\beta\in (\alpha-\min\{\gamma_{1},\pi-\gamma_{1}\},\alpha)$ and each sequence $(X_{j})\in t_{\beta}$, $$ \lim_{j\to \infty} \vec n_{f}(x_{j},y_{j}) = (\cos(\theta_{3}), \sin(\theta_{3}),0), $$ where $\theta_{3}= \alpha - \gamma_{1}+ \pi/2$. \item[(ii)] For each $\beta\in (\alpha-\min\{\gamma_{1},\pi-\gamma_{1}\},\alpha)$, and each sequence $(X_{j})\in t_{\beta}$, $$ \lim_{j\to \infty} \vec n_{f}(x_{j},y_{j}) = (\cos(\theta_{4}), \sin(\theta_{4}),0), $$ where $\theta_{4}= \alpha + \gamma_{1}+ \pi/2$. \end{itemize} \end{lemma} The proof of the lemma above is similar to that of Lemma 4 and uses items (e) of Lemmas \ref{lem6}-\ref{lem13} in \S 6 in place of items (a) of Lemmas \ref{lem6}-\ref{lem13}. \begin{theorem}[``Nonconvex Concus-Finn Conjecture'' with $\kappa=\lambda=0$] \label{thm6} Suppose \break $\alpha > \pi/2$ and $\kappa=\lambda=0$ in \eqref{capillary equation}. Suppose further that $(\gamma_{1},\gamma_{2})\in D^{+}_{1}\cup D^{-}_{1}\cup D^{+}_{2} \cup D^{-}_{2}$. Then every solution of \eqref{capillary equation}-\eqref{contactBC} must be discontinuous at $O=(0,0)$. \end{theorem} Using Lemmas \ref{lem1}-\ref{lem5}, we see that the proof of this theorem is the same as that of \cite[theorem 3.4]{JeffresLan1A}. \section{Proofs for nonconvex corners: Theorems \ref{thm4} and \ref{thm5}} In this section, we assume $\alpha> \pi/2$ and let $\mathcal{P}$ and $\mathcal{N}$ denote minimizers of $\Phi$ and $\Psi$ respectively. \begin{claim} \label{claim1} \rm Every component of $\Omega_{\infty}\cap \partial\mathcal{P}$ is unbounded and every component of $\Omega_{\infty}\cap \partial \mathcal{N}$ is unbounded. \end{claim} This claim is clear since if $L$ is a bounded component of $\Omega_{\infty}\cap \partial\mathcal{P}$ (or of $\Omega_{\infty}\cap \partial \mathcal{N}$), then $\partial L$ must be two distinct points of $\partial \Omega_{\infty}$ and $L\subset \Omega_{\infty}$; clearly this is impossible. \begin{claim} \label{claim2} \rm $\Omega_{\infty}\cap\partial \mathcal{ P}$ and $\Omega_{\infty}\cap\partial \mathcal{N}$ have at most two components. If $\Omega_{\infty}\cap\partial \mathcal{P}$ has a component $M$ which satisfies $\overline{M}\cap \partial\Omega_{\infty}=\emptyset$, then $\Omega_{\infty}\cap\partial \mathcal{P}$ has only this one component $M$. If $\Omega_{\infty}\cap\partial \mathcal{N}$ has a component $M$ which satisfies $\overline{M}\cap \partial\Omega_{\infty}=\emptyset$, then $\Omega_{\infty}\cap\partial \mathcal{N}=M$. \end{claim} \begin{proof} Using the arguments in \cite{JeffresLan1A}, \S 5, Claim \ref{claim2}, cases (c), (d), (g) and (h), we see that at most one component of $\Omega_{\infty}\cap \partial \mathcal{P}$ can have a point on $\Sigma_{1}$ in its closure. Using the arguments in \cite{JeffresLan1A}, \S 5, Claim \ref{claim2}, cases (e), (f), (k) and (l), we see that at most one component of $\Omega_{\infty}\cap \partial \mathcal{P}$ can have a point on $\Sigma_{2}$ in its closure. From \cite[Lemma 4.9]{JeffresLan1A}, we see that at most two components of $\Omega_{\infty}\cap \partial \mathcal{P}$ can have $O$ in their closure (and the measure of the angle between them is at least $\pi$). If there are two distinct components $L$ and $M$ of $\Omega_{\infty}\cap \partial \mathcal{P}$ with ${\overline L}\cap \partial \Omega_{\infty}=\emptyset$ and ${\overline M}\cap \partial \Omega_{\infty}=\emptyset$, then $L$ and $M$ must be parallel (e.g. \cite{JeffresLan1A}, \S 4, (i)-(ii)) and this violates \cite[Lemma 4.19]{JeffresLan1A}. Hence $\Omega_{\infty}\cap \partial \mathcal{P}$ has, at most, five components; let us say $\Omega_{\infty}\cap \partial \mathcal{P}=L_{1}\cup L_{2}\cup M_{1}\cup M_{1}\cup Q$, where $\overline{L_{1}}\cap \Sigma_{1}\neq\emptyset$, $\overline{L_{2}}\cap \Sigma_{2}\neq\emptyset$, $M_{1}=\{\theta=\beta_{1}\}$, $M_{2}=\{\theta=\beta_{2}\}$ (with $\beta_{1}-\beta_{2}\ge\pi$) and $\overline{Q}\cap \partial \Omega_{\infty}= \emptyset$. We shall show that the actual maximum number of components is two. Suppose two components of $\Omega_{\infty}\cap \partial \mathcal{ P}$ lie in $\{0\le\theta<\alpha\}$ and intersect $\overline{\Sigma_{1}}$. (In the notation of the previous paragraph, $\Omega_{\infty}\cap \partial \mathcal{P}$ contains $L_{1}$ and $M_{1}$ with $\beta_{1}\ge 0.$) The arguments in \cite{JeffresLan1A}, \S 5, Claim \ref{claim2}, cases (c), (d), (g) and (h) then yield a contradiction. Similarly, if two components of $\Omega_{\infty}\cap \partial \mathcal{P}$ lie in $\{-\alpha<\theta\le 0\}$ and intersect $\overline{\Sigma_{2}}$ (i.e. $\Omega_{\infty}\cap \partial \mathcal{P}$ contains $L_{2}$ and $M_{2}$ with $\beta_{1}\le 0$.) then \cite{JeffresLan1A}, \S 5, Claim \ref{claim2}, cases (e), (f), (k) and (l) yield a contradiction. Hence $\Omega_{\infty}\cap \partial \mathcal{P}$ can have at most two distinct components whose closures intersect $\partial \Omega_{\infty}$. The same conclusion holds for $\Omega_{\infty}\cap \partial \mathcal{N}$. Suppose $L$ and $M$ are components of $\Omega_{\infty}\cap \partial \mathcal{P}$ such that $L\cap \partial \Omega_{\infty}\neq \emptyset$ and $M\cap \partial \Omega_{\infty} = \emptyset$. This will result in a contradiction. From \cite[Lemma 4.19]{JeffresLan1A}, we see that $L$ and $M$ cannot be parallel. Let $L^{*}$ denote the line which contains $L$ and let $E$ denote the point of intersection of $L^{*}$ and $M$. We may suppose $\overline{L}\cap \overline{\Sigma_{1}}=\{A\}$; notice then that $E\in \{-\alpha<\theta\le 0\}$. The contradiction is obtained by modifying the proofs of \cite{JeffresLan1A}, \S 5, Claim \ref{claim2}, cases (g) and (h); we include the details here for the benefit of the reader. If $\Omega_{\infty}\cap \partial \mathcal{P}$ has only the two components $L$ and $M$, then $\partial \mathcal{P}= \Sigma_{2} \cup \overline{OA} \cup L \cup M$. If $\Omega_{\infty}\cap \partial \mathcal{P}$ has three components $L$, $M$ and $L_{2}$ (with $\overline{L_{2}}\cap \Sigma_{2}=\{Y\}$), then $\partial \mathcal{P}= L_{2} \cup \overline{OY} \cup \overline{OA} \cup L \cup M$; in this case, we note that if $L$ and $L_{2}$ are parallel, then $M$ is parallel to both $L$ and $L_{2}$ and this violates \cite[Lemma 4.19]{JeffresLan1A}. We exclude a potential third component $L_{2}$ of $\Omega_{\infty}\cap \partial \mathcal{P}$ in our arguments below since its inclusion would, at most, add a finite number of fixed terms to the right-hand sides of (\ref{ZZZ}) and (\ref{UUU}). \begin{figure}[ht] \begin{center} \begin{picture}(0,0) \includegraphics{fig4} % New2.pstex} \end{picture} \setlength{\unitlength}{1184sp} \begin{picture}(8038,8263)(2464,-9287) \put(5450,-8500){$E$} \put(4050,-4086){$A$} \put(4350,-6286){$B$} \put(7750,-3961){$C$} \put(3880,-2536){$D$} \put(5600,-5350){$O$} \end{picture} \end{center} \caption{Case (a)} \label{fig4} \end{figure} (a) Suppose $\mathcal{P}$ has only one component. Let $B$ be the point of intersection of $L^{*}$ and $\Sigma_{2}$ and let $D\in L$. Let $C$ be the orthogonal projection of $D$ on line $M$ and pick $T$ so that $T>\max\{OD,OC,OE\}$. Let $\Delta$ be the open, nonconvex polygon with boundary $OADCEB$. Since $\mathcal{P}$ minimizes $\Phi_{T}$, we have $\Phi_{T}(\mathcal{ P})\le \Phi_{T}(\mathcal{P}\setminus \Delta)$. Hence $$ EC + AD - \cos(\gamma_{1})OA - \cos(\gamma_{2})OB \le EB + CD. $$ Now $OA$, $OB$, $EB$ and $EA$ are fixed and $AD=ED-EA$; rewriting the inequality above yields the following inequality in which the right-hand side is fixed while the left-hand side goes to infinity as the length $ED$ goes to infinity: \begin{equation} \label{ZZZ} EC + ED - CD \le \cos(\gamma_{1})OA + \cos(\gamma_{2})OB + EB + EA, \end{equation} which is a contradiction. \begin{figure}[ht] \begin{center} \begin{picture}(0,0) \includegraphics{fig5} % New3.pstex \end{picture} \setlength{\unitlength}{1184sp} \begin{picture}(7896,8049)(2539,-9223) \put(5450,-8461){$E$} \put(4100,-4086){$A$} \put(4300,-6286){$B$} \put(7626,-3961){$C$} \put(3950,-2536){$D$} \put(5550,-5350){$O$} \end{picture} \end{center} \caption{Case (b)} \label{fig5} \end{figure} (b) Suppose $\mathcal{P}=\mathcal{P}_{1} \cup \mathcal{P}_{2}$, where $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are the disjoint, convex, open sets with boundaries $\overline{\Sigma_{1}} \cup L \setminus OA$ and $M$ respectively. Let $B$, $C$, $T$ and $\Delta$ be as in (a) above and let $D\in L$. Since $\mathcal{ P}$ minimizes $\Phi_{T}$, we have $\Phi_{T}(\mathcal{P})\le \Phi_{T}(\mathcal{P}\cup \Delta)$. Hence $$ EC + AD \le EB + CD - \cos(\gamma_{1})OA - \cos(\gamma_{2})OB. $$ Now $OA$, $OB$, $EB$ and $EA$ are fixed and $AD=ED-EA$; rewriting the inequality above yields the following inequality in which the right-hand side is fixed while the left-hand side goes to infinity as the length $ED$ goes to infinity: \begin{equation} \label{UUU} EC + ED - CD \le EB + EA- \cos(\gamma_{1})OA - \cos(\gamma_{2})OB, \end{equation} which is a contradiction. Since cases (a) and (b) and their counterparts when $\overline{L}\cap \overline{\Sigma_{2}} \neq \emptyset$ represent the only cases in which $\Omega_{\infty}\cap \partial \mathcal{ P}$ could have components $L$ and $M$ with $\overline{L}\cap \partial \Omega_{\infty}\neq \emptyset$ and $\overline{M}\cap \partial \Omega_{\infty} = \emptyset$, we see that if a component $M$ of $\Omega_{\infty}\cap \partial \mathcal{P}$ with $\overline{M}\cap \partial \Omega_{\infty} = \emptyset$ exists, then $\Omega_{\infty}\cap \partial \mathcal{P}$ has no other components. If no such component $M$ exists, then $\Omega_{\infty}\cap \partial \mathcal{P}$ could have two components $L_{1}$ and $L_{2}$ whose closures intersect $\overline{\Sigma_{1}}$ and $\overline{\Sigma_{2}}$ respectively, one component $L_{1}$ whose closure intersects $\partial \Omega_{\infty}$ or $\Omega_{\infty}\cap \partial \mathcal{P}$ could be empty. A similar argument for $\Omega_{\infty}\cap \partial \mathcal{N}$ completes the proof of the claim. \end{proof} \begin{claim} \label{claim3} \rm Suppose $M$ is a component of $\Omega_{\infty}\cap \partial \mathcal{P}$ and let $\omega$ denote the unit normal to $M$ in the direction of $\mathcal{P}$. Let $\sigma$ be the measure of the angle between $\omega$ and $\nu_{1}$. \begin{itemize} \item[(a)] If $\partial\Omega_{\infty}\subset \partial \mathcal{P}$, then $\sigma\ge \gamma_{1}$. \item[(b)] If $\partial\Omega_{\infty}\cap \partial \mathcal{ P}=\emptyset$, then $\sigma\le \gamma_{1}$. \vspace{2mm} \end{itemize} \end{claim} \begin{proof} Let $\Sigma_{1}^{*}$ denote the line which contains $\Sigma_{1}$ and let $C$ denote the point of intersection of $\Sigma_{1}^{*}$ and $M$. We will consider the proofs of (a) and (b) separately. \begin{figure}[ht] \begin{center} \begin{picture}(0,0) \includegraphics{fig6} %New5.pstex \end{picture} \setlength{\unitlength}{1184sp} \begin{picture}(7896,8625)(2389,-8273) \put(5600,-250){$B$} \put(3650,-2100){$\gamma_1$} \put(3950,-3580){$A$} \put(5700,-3700){$O$} \put(6556,-4380){$\sigma$} \put(7620,-5120){$C$} \put(6050,-5500){$\omega$} \put(6570,-5850){$\sigma$} \put(6700,-6300){$\nu_1$} \end{picture} \end{center} \caption{$\partial\Omega_{\infty}\subset \partial \mathcal{P}$} \label{fig6} \end{figure} Suppose (a) holds and $\sigma<\gamma_{1}$. Let $A\in\Sigma_{1}$ and pick $B\in M$ so that angle $OAB$ has measure $\pi-\gamma_{1}$. Notice that $\sigma$ is the measure of angle $ACB$. Since $\mathcal{P}$ minimizes $\Phi$, $$ \phi_{T}(\mathcal{P})\le \phi_{T}(\mathcal{P}\setminus \triangle ABC) $$ for $T$ large. Hence $$ BC-\cos(\gamma_{1})OA \le AB + OC $$ or $BC\le AB+\cos(\gamma_{1})AC+OC-\cos(\gamma_{1})OC$. If $\delta$ is the measure of angle $ABC$ (so $\delta=\gamma_{1}-\sigma$), then the law of sines implies $AC=(\sin(\delta)/\sin(\gamma_{1}))BC$ and $AB=(\sin(\sigma)/\sin(\gamma_{1})BC$. Hence $$ 1\le \cos(\gamma_{1}-\sigma) + (1-\cos(\gamma_{1}))\frac{OC}{BC}, $$ as a short calculation shows. For $BC$ sufficiently large, this yields a contradiction since $OC$ is fixed and $\gamma_{1}-\sigma>0$. \begin{figure}[ht] \begin{center} \begin{picture}(0,0) \includegraphics{fig7} %New6.pstex \end{picture} \setlength{\unitlength}{1184sp} \begin{picture}(7899,8625)(2389,-8273) \put(5500,-250){$B$} \put(3900,-3500){$A$} \put(4900,-2986){$\gamma_1$} \put(5650,-3800){$O$} \put(8501,-4661){$\omega$} \put(6701,-5511){$C$} \put(7750,-5800){$\sigma$} \put(6600,-6300){$\nu_1$} \end{picture} \end{center} \caption{$\partial\Omega_{\infty}\cap \partial \mathcal{P}=\emptyset$} \label{fig7} \end{figure} Suppose (b) holds and $\sigma>\gamma_{1}$. Let $A\in\Sigma_{1}$ and pick $B\in M$ so that angle $OAB$ has measure $\gamma_{1}$. Since $\mathcal{P}$ minimizes $\Phi$, $$ \phi_{T}(\mathcal{P})\le \phi_{T}(\mathcal{P}\cup \triangle ABC) $$ for $T$ large. Hence $$ BC \le AB - \cos(\gamma_{1})OA + OC $$ or $BC\le AB-\cos(\gamma_{1})AC+OC+\cos(\gamma_{1})OC$. If $\delta$ is the measure of angle $ABC$ (so $\delta=\sigma-\gamma_{1}$), then using the law of sines we obtain $$ 1\le -\cos(\pi+\gamma_{1}-\sigma) + (1+\cos(\gamma_{1}))\frac{OC}{BC}, $$ as a short calculation shows. For $BC$ sufficiently large, this yields a contradiction since $OC$ is fixed and $\pi+\gamma_{1}-\sigma<\pi$. \end{proof} The proofs of the following three claims are similar to the proof above. We leave the details to the reader. \begin{claim} \label{claim4} \rm Suppose $M$ is a component of $\Omega_{\infty}\cap \partial \mathcal{P}$ and let $\omega$ denote the unit normal to $M$ in the direction of $\mathcal{P}$. Let $\sigma$ be the measure of the angle between $\omega$ and $\nu_{2}$. \begin{itemize} \item[(a)] If $\partial\Omega_{\infty}\subset \partial \mathcal{P}$, then $\sigma\ge \gamma_{2}$. \item[(b)] If $\partial\Omega_{\infty}\cap \partial \mathcal{ P}=\emptyset$, then $\sigma\le \gamma_{2}$. \end{itemize} \end{claim} \begin{claim} \label{claim5} \rm Suppose $M$ is a component of $\Omega_{\infty}\cap \partial \mathcal{N}$ and let $\omega$ denote the unit normal to $M$ in the direction of $\Omega_{\infty}\setminus\mathcal{N}$. Let $\sigma$ be the measure of the angle between $\omega$ and $\nu_{1}$. \begin{itemize} \item[(a)] If $\partial\Omega_{\infty}\cap \partial \mathcal{ N}=\emptyset$, then $\sigma\ge \gamma_{1}$. \item[(b)] If $\partial\Omega_{\infty}\subset \partial \mathcal{N}$, then $\sigma\le \gamma_{1}$. \vspace{2mm} \end{itemize} \end{claim} \begin{claim} \label{claim6} \rm Suppose $M$ is a component of $\Omega_{\infty}\cap \partial \mathcal{N}$ and let $\omega$ denote the unit normal to $M$ in the direction of $\Omega_{\infty}\setminus\mathcal{N}$. Let $\sigma$ be the measure of the angle between $\omega$ and $\nu_{2}$. \begin{itemize} \item[(a)] If $\partial\Omega_{\infty}\cap \partial \mathcal{ N}=\emptyset$, $\sigma\ge \gamma_{2}$. \item[(b)] If $\partial\Omega_{\infty}\subset \partial \mathcal{N}$, then $\sigma\le \gamma_{2}$. \end{itemize} \end{claim} \subsection*{Proof of Theorems \ref{thm4} and \ref{thm5}} Consider first the case that $\Omega_{\infty}\cap \partial \mathcal{P}$ has exactly one component, denoted by $L$. Then one of the following holds: $$ \overline{L}\cap \Sigma_{1}\neq \emptyset, \quad \overline{L}\cap \Sigma_{2}\neq\emptyset,\quad \overline{L}\cap \partial\Omega_{\infty}=\{O\}\quad\text{or} \quad \overline{L}\cap \partial\Omega_{\infty}=\emptyset. $$ Suppose $\overline{L}\cap \Sigma_{1}\neq\emptyset$ and let $A$ be the point of intersection of $\overline{L}$ and $\Sigma_{1}$. If $OA\in \overline{\Omega_{\infty}\setminus \mathcal{P}}$, then \cite[Lemma 4.8]{JeffresLan1A}, implies $\Sigma_{1}\setminus OA$ and $L$ meet at $A$ in an angle of measure $\gamma_{1}$ and a slight modification of the argument of the proof of \cite[Lemma 4.6]{JeffresLan1A}, implies $\gamma_{1}+\pi-\gamma_{2} \le 2\alpha$. If $OA\in \overline{\mathcal{P}}$, then \cite[Lemma 4.8]{JeffresLan1A}, implies $OA$ and $L$ meet at $A$ in an angle of measure $\gamma_{1}$ and a slight modification of the argument of the proof of \cite[Lemma 4.7]{JeffresLan1A}, implies $\pi-\gamma_{1}+\gamma_{2} \le 2\alpha$. Hence either (ii) or (iii) of Theorem \ref{thm4} holds. If $\overline{L}\cap \Sigma_{2}\neq\emptyset$, then \cite[Lemmas 4.6, 4.7, 4.10]{JeffresLan1A} imply that either (iv) or (v) of Theorem \ref{thm4} holds. If $\overline{L}\cap \partial\Omega_{\infty}=\{O\}$, then either $\Sigma_{1}\subset \overline{\mathcal{P}}$ and so (vi) of Theorem \ref{thm4} holds (by \cite[Lemma 4.6, 4.11]{JeffresLan1A}) or $\Sigma_{2}\subset \overline{\mathcal{P}}$ and so (vii) of Theorem \ref{thm4} holds (by \cite[Lemma 4.7, 4.9]{JeffresLan1A}) \begin{figure}[ht] \begin{center} \begin{picture}(0,0) \includegraphics{fig8} % New4.pstex \end{picture} \setlength{\unitlength}{1184sp} \begin{picture}(5796,6761)(2389,-7304) \put(5800,-2986){$B$} \put(5250,-3700){$\delta$} \put(3751,-4100){$O$} \put(5750,-5050){$\beta$} \put(6676,-6000){$A$} \end{picture} \end{center} \caption{Cases (ix) and (xii)} \label{fig8} \end{figure} Suppose $\overline{L}\cap \partial\Omega_{\infty}=\emptyset$. Then either $\partial\Omega_{\infty}\subset \overline{\mathcal{P}}$ or $\partial\Omega_{\infty}\cap \overline{\mathcal{P}}=\emptyset$. Let $\Sigma^{*}_{1}$ and $\Sigma^{*}_{2}$ be the lines on which $\Sigma_{1}$ and $\Sigma_{2}$ respectively lie. Now \cite[Lemma 4.20]{JeffresLan1A} implies $L$ is not parallel to either $\Sigma^{*}_{1}$ or $\Sigma^{*}_{2}$. Let $A,B \in \Omega_{\infty}$ satisfy $\Sigma^{*}_{1}\cap L=\{A\}$ and $\Sigma^{*}_{2}\cap L=\{B\}$. Let $\beta$ and $\delta$ be the measures of the angles $OAB$ and $OBA$ respectively. Let $\omega$ denote the unit normal to $L$ in the direction of $\mathcal{P}$. If $\partial\Omega_{\infty}\subset \overline{\mathcal{P}}$, then $\beta$ is the measure of the angle between $\omega$ and $\nu_{1}$, $\delta$ is the measure of the angle between $\omega$ and $\nu_{2}$, (a) of Claims \ref{claim3} and \ref{claim4} imply $\beta\ge \gamma_{1}$ and $\delta\ge \gamma_{2}$ and so (xii) of Theorem \ref{thm4} holds. If $\partial\Omega_{\infty}\cap \overline{\mathcal{P}}=\emptyset$, then $\beta$ is the measure of the angle between $-\omega$ and $\nu_{1}$, $\delta$ is the measure of the angle between $-\omega$ and $\nu_{2}$, (b) of Claims \ref{claim3} and \ref{claim4} imply $\beta\ge \pi-\gamma_{1}$ and $\delta\ge \pi-\gamma_{2}$ and so (ix) of Theorem \ref{thm4} holds. Consider next the case that $\Omega_{\infty}\cap \partial \mathcal{P}$ has exactly two components, denoted by $L$ and $M$ with $\overline{L}\cap \overline{\Sigma_{1}}=\{A\}$ and $\overline{M}\cap \overline{\Sigma_{2}}=\{B\}$. Then the following combinations are possible: \begin{itemize} \item[(a)] $A\in \Sigma_{1}$ and $B\in \Sigma_{2}$; \item[(b)] $A=O$ and $B\in \Sigma_{2}$; \item[(c)] $A\in \Sigma_{1}$ and $B=O$; \item[(d)] $A=O$ and $B=O$. \end{itemize} If (a) holds, then (xiv) and (xv) of Theorem \ref{thm4} follow from \cite[Lemma 4.8, 4.10]{JeffresLan1A}. If (b) holds, then (xviii) and (xix) of Theorem \ref{thm4} follow from \cite[Lemmas 4.6, 4.9, 4.10]{JeffresLan1A}. If (c) holds, then (xvi) and (xvii) of Theorem \ref{thm4} follow from \cite[Lemmas 4.7, 4.8, 4.11]{JeffresLan1A}. If (d) holds, then (x) and (xiii) of Theorem \ref{thm4} follow from \cite[Lemma 4.6, 4.7, 4.9, 4.11]{JeffresLan1A}; we note that (viii) and (xi) are special cases of (x) and (xiii) respectively. This completes the proof of Theorem \ref{thm4}. The proof of \cite[Theorem 2.2]{JeffresLan1A} follows by similar arguments. \section{Some Additional Corollaries} The proofs of the following corollaries are simple exercises in checking angle conditions in Theorems \ref{thm4} and \ref{thm5}. \begin{corollary} \label{coro5} Suppose $\alpha>\pi/2$, $\gamma_{1}+\gamma_{2}<2\alpha-\pi$, $\alpha-(\pi-\gamma_{1}) \ge -\alpha+(\pi-\gamma_{2})$, $\gamma_{1}\le \pi/2$ and $\gamma_{2}\le \pi/2$. Let $r_{0}>0$, $\beta\in (-\alpha,\alpha)$ and $Y=(r_{0}\cos(\beta),r_{0}\sin(\beta))$ and suppose $Y\in \partial \mathcal{P} \cap \partial \mathcal{N}$. \begin{itemize} \item[(a)] If $-\alpha<\beta<-\alpha+\gamma_{2}$, then one of cases (iv), (v), (xiv) or (xviii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(b)] If $-\alpha+\gamma_{2} \le \beta<-\alpha+\pi-\gamma_{2}$, then one of cases (iv), (vii), (xi), (xii), (xiii) or (xvi) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(c)] If $-\alpha+\pi-\gamma_{2}\le \beta\le \alpha-(\pi-\gamma_{1})$, then one of cases (vi), (vii) or (xii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(d)] If $\alpha-(\pi-\gamma_{1})<\beta\le \alpha-\gamma_{1}$, then one of cases (iii), (vi), (xi), (xii) ,(xiii) or (xviii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(e)] If $\alpha-\gamma_{1}<\beta<\alpha$, then one of cases (ii), (iii), (xiv) or (xvi) of Theorems \ref{thm4} and \ref{thm5} holds. \end{itemize} \end{corollary} \begin{corollary} \label{coro6} Suppose $\alpha>\pi/2$, $\gamma_{1}+\gamma_{2}<2\alpha-\pi$, $\alpha-(\pi-\gamma_{1}) < -\alpha+(\pi-\gamma_{2})$, $\gamma_{1}\le \pi/2$ and $\gamma_{2}\le \pi/2$. Let $r_{0}>0$, $\beta\in (-\alpha,\alpha)$ and $Y=(r_{0}\cos(\beta),r_{0}\sin(\beta))$ and suppose $Y\in \partial \mathcal{P} \cap \partial \mathcal{N}$. \begin{itemize} \item[(a)] If $-\alpha<\beta<-\alpha+\gamma_{2}$, then one of cases (iv), (v), (xiv) or (xviii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(b)] If $-\alpha+\gamma_{2} \le \beta<\alpha-(\pi-\gamma_{1})$, then one of cases (iv), (vii), (xi), (xii), (xiii) or (xvi) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(c)] If $\alpha-(\pi-\gamma_{1}) \le \beta\le -\alpha+(\pi-\gamma_{2})$, then one of cases (iii), (iv) or (xii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(d)] If $-\alpha+(\pi-\gamma_{2})<\beta\le \alpha-\gamma_{1}$, then one of cases (iii), (vi), (xi), (xii), (xiii) or (xviii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(e)] If $\alpha-\gamma_{1}<\beta<\alpha$, then one of cases (ii), (iii), (xiv) or (xvi) of Theorems \ref{thm4} and \ref{thm5} holds. \end{itemize} \end{corollary} \begin{corollary} \label{coro7} Suppose $\alpha>\pi/2$, $\gamma_{1}+\gamma_{2}<2\alpha-\pi$, $\gamma_{1} > \pi/2$ and $\gamma_{2}\le \pi/2$. Let $r_{0}>0$, $\beta\in (-\alpha,\alpha)$ and $Y=(r_{0}\cos(\beta),r_{0}\sin(\beta))$ and suppose $Y\in \partial \mathcal{P} \cap \partial \mathcal{N}$. \begin{itemize} \item[(a)] If $-\alpha<\beta<-\alpha+\gamma_{2}$, then one of cases (iv), (v), (xiv) or (xviii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(b)] If $-\alpha+\gamma_{2} \le \beta<-\alpha+(\pi-\gamma_{2})$, then one of cases (iv), (vii), (xi), (xii), (xiii) or (xvi) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(c)] If $-\alpha+(\pi-\gamma_{2}) \le \beta\le \alpha-\gamma_{1}$, then one of cases (vi), (vii) or (xii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(d)] If $\alpha-\gamma_{1}<\beta\le \alpha-(\pi-\gamma_{1})$, then one of cases (ii), (vii), (xiv) or (xvi) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(e)] If $\alpha-(\pi-\gamma_{1})<\beta<\alpha$, then one of cases (ii), (iii), (xiv) or (xvi) of Theorems \ref{thm4} and \ref{thm5} holds. \end{itemize} \end{corollary} \begin{corollary} \label{coro8} Suppose $\alpha>\pi/2$, $\gamma_{1}+\gamma_{2}<2\alpha-\pi$, $\gamma_{1} \le \pi/2$ and $\gamma_{2} > \pi/2$. Let $r_{0}>0$, $\beta\in (-\alpha,\alpha)$ and $Y=(r_{0}\cos(\beta),r_{0}\sin(\beta))$ and suppose $Y\in \partial \mathcal{P} \cap \partial \mathcal{N}$. \begin{itemize} \item[(a)] If $-\alpha<\beta<-\alpha+(\pi-\gamma_{2})$, then one of cases (iv), (v), (xiv) or (xviii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(b)] If $-\alpha+(\pi-\gamma_{2}) \le \beta<-\alpha+\gamma_{2}$, then one of cases (v), (vi), (xiv) or (xviii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(c)] If $-\alpha+\gamma_{2} \le \beta\le \alpha-(\pi-\gamma_{1})$, then one of cases (vi), (vii) or (xii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(d)] If $\alpha-(\pi-\gamma_{1})<\beta\le \alpha-\gamma_{1}$, then one of cases (iii), (vi), (xi), (xii), (xiii) or (xviii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(e)] If $\alpha-\gamma_{1}<\beta<\alpha$, then one of cases (ii), (iii), (xiv) or (xvi) of Theorems \ref{thm4} and \ref{thm5} holds. \end{itemize} \end{corollary} \begin{corollary} \label{coro9} Suppose $\alpha>\pi/2$, $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$, $\alpha-\gamma_{1} \ge -\alpha+\gamma_{2}$, $\gamma_{1}\ge \pi/2$ and $\gamma_{2}\ge \pi/2$. Let $r_{0}>0$, $\beta\in (-\alpha,\alpha)$ and $Y=(r_{0}\cos(\beta),r_{0}\sin(\beta))$ and suppose $Y\in \partial \mathcal{P} \cap \partial \mathcal{N}$. \begin{itemize} \item[(a)] If $-\alpha<\beta<-\alpha+(\pi-\gamma_{2})$, then one of cases (iv), (v), (xv) or (xix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(b)] If $-\alpha+(\pi-\gamma_{2}) \le \beta<-\alpha+\gamma_{2}$, then one of cases (v), (vi), (viii), (ix), (x) or (xvii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(c)] If $-\alpha+\gamma_{2}\le \beta\le \alpha-\gamma_{1}$, then one of cases (vi), (vii) or (ix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(d)] If $\alpha-\gamma_{1}<\beta\le \alpha-(\pi-\gamma_{1})$, then one of cases (ii), (vii), (viii), (ix), (x) or (xix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(e)] If $\alpha-(\pi-\gamma_{1})<\beta<\alpha$, then one of cases (ii), (iii), (xv) or (xvii) of Theorems \ref{thm4} and \ref{thm5} holds. \end{itemize} \end{corollary} \begin{corollary} \label{coro10} Suppose $\alpha>\pi/2$, $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$, $\alpha-\gamma_{1} < -\alpha+\gamma_{2}$, $\gamma_{1}\ge \pi/2$ and $\gamma_{2}\ge \pi/2$. Let $r_{0}>0$, $\beta\in (-\alpha,\alpha)$ and $Y=(r_{0}\cos(\beta),r_{0}\sin(\beta))$ and suppose $Y\in \partial \mathcal{P} \cap \partial \mathcal{N}$. \begin{itemize} \item[(a)] If $-\alpha<\beta<-\alpha+(\pi-\gamma_{2})$, then one of cases (iv), (v), (xv) or (xix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(b)] If $-\alpha+(\pi-\gamma_{2}) \le \beta<\alpha-\gamma_{1}$, then one of cases (v), (vi), (viii), (ix), (x) or (xvii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(c)] If $\alpha-\gamma_{1}\le \beta\le -\alpha+\gamma_{2}$, then one of cases (ii), (v) or (ix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(d)] If $-\alpha+\gamma_{2}<\beta\le \alpha-(\pi-\gamma_{1})$, then one of cases (ii), (vii), (viii), (ix), (x) or (xix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(e)] If $\alpha-(\pi-\gamma_{1})<\beta<\alpha$, then one of cases (ii), (iii), (xv) or (xvii) of Theorems \ref{thm4} and \ref{thm5} holds. \end{itemize} \end{corollary} \begin{corollary} \label{coro11} Suppose $\alpha>\pi/2$, $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$, $\gamma_{1} < \pi/2$ and $\gamma_{2}\ge \pi/2$. Let $r_{0}>0$, $\beta\in (-\alpha,\alpha)$ and $Y=(r_{0}\cos(\beta),r_{0}\sin(\beta))$ and suppose $Y\in \partial \mathcal{P} \cap \partial \mathcal{N}$. \begin{itemize} \item[(a)] If $-\alpha<\beta<-\alpha+(\pi-\gamma_{2})$, then one of cases (iv), (v), (xv) or (xix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(b)] If $-\alpha+(\pi-\gamma_{2}) \le \beta<-\alpha+\gamma_{2}$, then one of cases (v), (vi), (viii), (ix), (x) or (xvii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(c)] If $-\alpha+\gamma_{2}\le \beta\le \alpha-(\pi-\gamma_{1})$, then one of cases (vi), (vii) or (ix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(d)] If $\alpha-(\pi-\gamma_{1})<\beta\le \alpha-\gamma_{1}$, then one of cases (iii), (vi), (xv) or (xvii) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(e)] If $\alpha-\gamma_{1}<\beta<\alpha$, then one of cases (ii), (iii), (xv) or (xvii) of Theorems \ref{thm4} and \ref{thm5} holds. \end{itemize} \end{corollary} \begin{corollary} \label{cor12} Suppose $\alpha>\pi/2$, $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$, $\gamma_{1}\ge \pi/2$ and $\gamma_{2} < \pi/2$. Let $r_{0}>0$, $\beta\in (-\alpha,\alpha)$ and $Y=(r_{0}\cos(\beta),r_{0}\sin(\beta))$ and suppose $Y\in \partial \mathcal{P} \cap \partial \mathcal{N}$. \begin{itemize} \item[(a)] If $-\alpha<\beta<-\alpha+\gamma_{2}$, then one of cases (iv), (v), (xv) or (xix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(b)] If $-\alpha+\gamma_{2} \le \beta<-\alpha+(\pi-\gamma_{2})$, then one of cases (iv), (vii), (xv) and (xix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(c)] If $-\alpha+(\pi-\gamma_{2})\le \beta\le \alpha-\gamma_{1}$, then one of cases (vi), (vii) or (ix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(d)] If $\alpha-\gamma_{1}<\beta\le \alpha-(\pi-\gamma_{1})$, then one of cases (ii), (vii), (viii), (ix), (x) or (xix) of Theorems \ref{thm4} and \ref{thm5} holds. \item[(e)] If $\alpha-(\pi-\gamma_{1})<\beta<\alpha$, then one of cases (ii), (iii), (xv) or (xvii) of Theorems \ref{thm4} and \ref{thm5} holds. \end{itemize} \end{corollary} \begin{remark} \label{rmk2} \rm If $2\alpha<\gamma_{1}+\gamma_{2}+\pi$, then cases (xi), (xii) and (xiii) cannot occur. If $2\alpha+\gamma_{1}+\gamma_{2}<3\pi$, then cases (viii), (ix) and (x) cannot occur. \end{remark} \section{Lemmas \ref{lem6}-\ref{lem13}} The following lemmas are used in the proofs of Lemmas \ref{lem4} and \ref{lem5}. Let us recall that we defined $$ C_{\beta}(f)=\{ \eta\in S^{2} :\eta=\lim_{j\to\infty} \vec n_{f}(X_{j}) \text{ for some }(X_{j})\in t_{\beta} \} $$ for each $\beta\in (-\alpha,\alpha)$ and set $C(f)=\cup_{\beta\in (-\alpha,\alpha)} C_{\beta}(f)$. \begin{lemma} \label{lem6} Suppose $\alpha > \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{+}_{1}$ (i.e. $\gamma_{1}+\gamma_{2}<2\alpha-\pi$). Suppose further that $\alpha-(\pi-\gamma_{1}) \ge -\alpha+(\pi-\gamma_{2})$, $\gamma_{1}\le \pi/2$ and $\gamma_{2}\le \pi/2$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC} and define $\vec n(x,y)$ by (\ref{rabbit}). Then: \begin{itemize} \item[(a)] If $\beta\in (-\alpha,-\alpha+\gamma_{2})$ and $\eta\in C_{\beta}(f)$, then $\eta$ is one of the following: $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$. \item[(b)] If $\beta\in [-\alpha+\gamma_{2},-\alpha+(\pi-\gamma_{2}))$ and $\eta\in C_{\beta}(f)$, then $\eta$ is one of the following: $(\cos(\beta-\frac{\pi}{2}), \sin(\beta-\frac{\pi}{2}),0)$, $(\cos(\omega), \sin(\omega),0)$ or $(\cos(\theta), \sin(\theta),0)$ for some $\theta\in [2\pi-\alpha+\gamma_{2}-\frac{\pi}{2},\alpha-\gamma_{1} +\frac{\pi}{2}]$, where $\omega=\frac{3\pi}{2} - \alpha - \gamma_{2}$. \item[(c)] If $\beta\in [-\alpha+(\pi-\gamma_{2}),\alpha-(\pi-\gamma_{1})]$ and $\eta\in C_{\beta}(f)$, then $\eta$ is one of the following: $(-\sin(\beta),\cos(\beta),0)$, $(\sin(\beta), -\cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(d)] If $\beta\in (\alpha-(\pi-\gamma_{1}),\alpha-\gamma_{1}]$ and $\eta\in C_{\beta}(f)$, then $\eta$ is one of the following: $(-\sin(\beta),\cos(\beta),0)$, $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(e)] If $\beta\in (\alpha-\gamma_{1},\alpha)$ and $\eta\in C_{\beta}(f)$, then $\eta$ is one of the following: $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$ or $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$. \end{itemize} \end{lemma} \begin{proof} Let $\beta\in (-\alpha,\alpha)$ and $\eta\in C_{\beta}(f)$. Let $\{(x_{j},y_{j}): j\in \mathbb{N} \}$ be a sequence in $\Omega$ satisfying (\ref{krogoth}) such that $\vec n_{f}(x_{j},y_{j}) \to \eta$ as $j \to \infty$. For each $j\in \mathbb{N}$, define $f_{j}\in C^{2}(\Omega_{j})\cap C^{1}(\overline{\Omega_{j}}\setminus \{O\})$ by $$ f_{j}(x,y)=\frac{f(\epsilon_{j}x,\epsilon_{j}y)-f(x_{j},y_{j})} {\epsilon_{j}}, $$ where $\epsilon_{j}=\sqrt{x_{j}^2+y_{j}^2}$. Let $U_{j}=\{(x,t)\in \Omega_{j}\times \mathbb{R}:t \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{+}_{1}$ (i.e. $\gamma_{1}+\gamma_{2}<2\alpha-\pi$). Suppose further that $\alpha-(\pi-\gamma_{1}) < -\alpha+(\pi-\gamma_{2})$, $\gamma_{1}\le \pi/2$ and $\gamma_{2}\le \pi/2$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC} and define $\vec n(x,y)$ by (\ref{rabbit}). For each $\beta\in (-\alpha,\alpha)$, there exists a unit vector $\vec n_{\beta}=\vec n_{\beta}(f)$ such that if $\{ (x_{j},y_{j})\}$ is any sequence in $\Omega$ satisfying $\lim_{j\to\infty} (x_{j},y_{j})=(0,0)$ and (\ref{krogoth}), then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = \vec n_{\beta}$. In addition: \begin{itemize} \item[(a)] If $\beta\in (-\alpha,-\alpha+\gamma_{2})$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$. \item[(b)] If $\beta\in [-\alpha+\gamma_{2},\alpha-(\pi-\gamma_{1}))$, then $\vec n_{\beta}$ is one of the following:\\ $(\sin(\beta), -\cos(\beta),0)$, $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(c)] If $\beta\in [\alpha-(\pi-\gamma_{1}),-\alpha+(\pi-\gamma_{2})]$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha+\gamma_{1}),\cos(\alpha+\gamma_{1}),0)$, $(-\sin(\alpha+\gamma_{2}),-\cos(\alpha+\gamma_{2}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(d)] If $\beta\in (-\alpha+(\pi-\gamma_{2}),\alpha-\gamma_{1}]$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\beta),\cos(\beta),0)$, $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(e)] If $\beta\in (\alpha-\gamma_{1},\alpha)$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$ or $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$. \end{itemize} \end{lemma} \begin{lemma} \label{lem8} Suppose $\alpha > \pi/2$, $(\gamma_{1},\gamma_{2})$ lies in $D^{+}_{1}$ (i.e. $\gamma_{1}+\gamma_{2}<2\alpha-\pi$), $\gamma_{1} > \pi/2$ and $\gamma_{2}\le \pi/2$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC} and define $\vec n(x,y)$ by (\ref{rabbit}). For each $\beta\in (-\alpha,\alpha)$, there exists a unit vector $\vec n_{\beta}=\vec n_{\beta}(f)$ such that if $\{ (x_{j},y_{j})\}$ is any sequence in $\Omega$ satisfying $\lim_{j\to\infty} (x_{j},y_{j})=(0,0)$ and (\ref{krogoth}), then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = \vec n_{\beta}$. In addition: \begin{itemize} \item[(a)] If $\beta\in (-\alpha,-\alpha+\gamma_{2})$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$. \item[(b)] If $\beta\in [-\alpha+\gamma_{2},-\alpha+(\pi-\gamma_{2}))$, then $\vec n_{\beta}$ is one of the following:\\ $(\sin(\beta), -\cos(\beta),0)$, $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(c)] If $\beta\in [-\alpha+(\pi-\gamma_{2}),\alpha-\gamma_{1}]$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\beta),\cos(\beta),0)$, $(\sin(\beta), -\cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(d)] If $\beta\in (\alpha-\gamma_{1},\alpha-(\pi-\gamma_{1})]$, then $\vec n_{\beta}$ is one of the following:\\ $(\sin(\beta),-\cos(\beta),0)$ or $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$. \item[(e)] If $\beta\in (\alpha-(\pi-\gamma_{1}),\alpha)$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$ or $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$. \end{itemize} \end{lemma} \begin{lemma} \label{lem9} Suppose $\alpha > \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{+}_{1}$ (i.e. $\gamma_{1}+\gamma_{2}<2\alpha-\pi$). Suppose further that $\alpha-(\pi-\gamma_{1}) \ge -\alpha+(\pi-\gamma_{2})$, $\gamma_{1}\le \pi/2$ and $\gamma_{2}\le \pi/2$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC} and define $\vec n(x,y)$ by (\ref{rabbit}). For each $\beta\in (-\alpha,\alpha)$, there exists a unit vector $\vec n_{\beta}=\vec n_{\beta}(f)$ such that if $\{ (x_{j},y_{j})\}$ is any sequence in $\Omega$ satisfying $\lim_{j\to\infty} (x_{j},y_{j})=(0,0)$ and (\ref{krogoth}), then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = \vec n_{\beta}$. In addition: \begin{itemize} \item[(a)] If $\beta\in (-\alpha,-\alpha+(\pi-\gamma_{2}))$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$. \item[(b)] If $\beta\in [-\alpha+(\pi-\gamma_{2}),-\alpha+\gamma_{2})$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\beta), \cos(\beta),0)$ or $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$. \item[(c)] If $\beta\in [-\alpha+\gamma_{2},\alpha-(\pi-\gamma_{1})]$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\beta),\cos(\beta),0)$, $(\sin(\beta), -\cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(d)] If $\beta\in (\alpha-(\pi-\gamma_{1}),\alpha-\gamma_{1}]$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\beta),\cos(\beta),0)$, $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(e)] If $\beta\in (\alpha-\gamma_{1},\alpha)$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$ or $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$. \end{itemize} \end{lemma} \begin{lemma} \label{lem10} Suppose $\alpha > \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{-}_{1}$ (i.e. $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$). Suppose further that $\alpha-\gamma_{1} \ge -\alpha-\gamma_{2}$, $\gamma_{1}\ge \pi/2$ and $\gamma_{2}\ge \pi/2$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC} and define $\vec n(x,y)$ by (\ref{rabbit}). For each $\beta\in (-\alpha,\alpha)$, there exists a unit vector $\vec n_{\beta}=\vec n_{\beta}(f)$ such that if $\{ (x_{j},y_{j})\}$ is any sequence in $\Omega$ satisfying $\lim_{j\to\infty} (x_{j},y_{j})=(0,0)$ and (\ref{krogoth}), then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = \vec n_{\beta}$. In addition: \begin{itemize} \item[(a)] If $\beta\in (-\alpha,-\alpha+(\pi-\gamma_{2}))$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$. \item[(b)] If $\beta\in [-\alpha+(\pi-\gamma_{2}),-\alpha+\gamma_{2})$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$, $(-\sin(\beta), \cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(c)] If $\beta\in [-\alpha+\gamma_{2},\alpha-\gamma_{1}]$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\beta),\cos(\beta),0)$, $(\sin(\beta), -\cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(d)] If $\beta\in (\alpha-\gamma_{1},\alpha-(\pi-\gamma_{1})]$, then $\vec n_{\beta}$ is one of the following:\\ $(\sin(\beta),-\cos(\beta),0)$, $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(e)] If $\beta\in (\alpha-(\pi-\gamma_{1}),\alpha)$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$ or $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$. \end{itemize} \end{lemma} \begin{lemma} \label{lem11} Suppose $\alpha > \pi/2$ and $(\gamma_{1},\gamma_{2})$ lies in $D^{-}_{1}$ (i.e. $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$). Suppose further that $\alpha-\gamma_{1} < -\alpha-\gamma_{2}$, $\gamma_{1}\ge \pi/2$ and $\gamma_{2}\ge \pi/2$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC} and define $\vec n(x,y)$ by (\ref{rabbit}). For each $\beta\in (-\alpha,\alpha)$, there exists a unit vector $\vec n_{\beta}=\vec n_{\beta}(f)$ such that if $\{ (x_{j},y_{j})\}$ is any sequence in $\Omega$ satisfying $\lim_{j\to\infty} (x_{j},y_{j})=(0,0)$ and (\ref{krogoth}), then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = \vec n_{\beta}$. In addition: \begin{itemize} \item[(a)] If $\beta\in (-\alpha,-\alpha+(\pi-\gamma_{2}))$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$. \item[(b)] If $\beta\in [-\alpha+(\pi-\gamma_{2}),\alpha-\gamma_{1})$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$, $(-\sin(\beta), \cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(c)] If $\beta\in [\alpha-\gamma_{1},-\alpha+\gamma_{2}]$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$, $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(d)] If $\beta\in (-\alpha+\gamma_{2},\alpha-(\pi-\gamma_{1})]$, then $\vec n_{\beta}$ is one of the following:\\ $(\sin(\beta),-\cos(\beta),0)$, $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(e)] If $\beta\in (\alpha-(\pi-\gamma_{1}),\alpha)$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$ or $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$. \end{itemize} \end{lemma} \begin{lemma} \label{lem12} Suppose $\alpha > \pi/2$, $(\gamma_{1},\gamma_{2})$ lies in $D^{-}_{1}$ (i.e. $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$). $\gamma_{1} < \pi/2$ and $\gamma_{2}\ge \pi/2$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC} and define $\vec n(x,y)$ by (\ref{rabbit}). For each $\beta\in (-\alpha,\alpha)$, there exists a unit vector $\vec n_{\beta}=\vec n_{\beta}(f)$ such that if $\{ (x_{j},y_{j})\}$ is any sequence in $\Omega$ satisfying $\lim_{j\to\infty} (x_{j},y_{j})=(0,0)$ and (\ref{krogoth}), then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = \vec n_{\beta}$. In addition: \begin{itemize} \item[(a)] If $\beta\in (-\alpha,-\alpha+(\pi-\gamma_{2}))$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$. \item[(b)] If $\beta\in [-\alpha+(\pi-\gamma_{2}),-\alpha+\gamma_{2})$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$, $(-\sin(\beta), \cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(c)] If $\beta\in [-\alpha+\gamma_{2},\alpha-(\pi-\gamma_{1})]$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\beta),\cos(\beta),0)$, $(\sin(\beta), -\cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(d)] If $\beta\in (\alpha-(\pi-\gamma_{1}),\alpha-\gamma_{1}]$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$ or $(-\sin(\beta),\cos(\beta),0)$. \item[(e)] If $\beta\in (\alpha-\gamma_{1},\alpha)$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$ or $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$. \end{itemize} \end{lemma} \begin{lemma} \label{lem13} Suppose $\alpha > \pi/2$, $(\gamma_{1},\gamma_{2})$ lies in $D^{-}_{1}$ (i.e. $\gamma_{1}+\gamma_{2}>3\pi-2\alpha$), $\gamma_{1}\ge \pi/2$ and $\gamma_{2} < \pi/2$. Let $f\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega}\setminus \{O\})$ satisfy \eqref{capillary equation} and \eqref{contactBC} and define $\vec n(x,y)$ by (\ref{rabbit}). For each $\beta\in (-\alpha,\alpha)$, there exists a unit vector $\vec n_{\beta}=\vec n_{\beta}(f)$ such that if $\{ (x_{j},y_{j})\}$ is any sequence in $\Omega$ satisfying $\lim_{j\to\infty} (x_{j},y_{j})=(0,0)$ and (\ref{krogoth}), then $\lim_{j\to \infty} \vec n(x_{j},y_{j}) = \vec n_{\beta}$. In addition: \begin{itemize} \item[(a)] If $\beta\in (-\alpha,-\alpha+\gamma_{2})$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\alpha-\gamma_{2}), -\cos(\alpha-\gamma_{2}),0)$ or $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$. \item[(b)] If $\beta\in [-\alpha+\gamma_{2},-\alpha+(\pi-\gamma_{2}))$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha+\gamma_{2}), -\cos(\alpha+\gamma_{2}),0)$ or $(\sin(\beta), -\cos(\beta),0)$. \item[(c)] If $\beta\in [-\alpha+(\pi-\gamma_{2}),\alpha-\gamma_{1}]$, then $\vec n_{\beta}$ is one of the following:\\ $(-\sin(\beta),\cos(\beta),0)$, $(\sin(\beta), -\cos(\beta),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(d)] If $\beta\in (\alpha-\gamma_{1},\alpha-(\pi-\gamma_{1})]$, then $\vec n_{\beta}$ is one of the following:\\ $(\sin(\beta),-\cos(\beta),0)$, $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$ or $(\sin(\theta), -\cos(\theta),0)$ for some $\theta$ with $\alpha+\pi-\gamma_{1}\le \theta \le 2\pi-\alpha+\gamma_{2}$. \item[(e)] If $\beta\in (\alpha-(\pi-\gamma_{1}),\alpha)$, then $\vec n_{\beta}$ is one of the following: $(-\sin(\alpha-\gamma_{1}), \cos(\alpha-\gamma_{1}),0)$ or $(-\sin(\alpha+\gamma_{1}), \cos(\alpha+\gamma_{1}),0)$. \end{itemize} \end{lemma} \begin{thebibliography}{00} \bibitem{BearHile} H. 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