\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 65, pp. 1--37.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/65\hfil Local solvability] {Local solvability of degenerate Monge-Amp\`{e}re equations and applications to geometry} \author[M. A. Khuri\hfil EJDE-2007/65\hfilneg] {Marcus A. Khuri} \address{Marcus A. Khuri \newline Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA} \email{khuri@math.sunysb.edu} \thanks{Submitted February 28, 2007. Published May 9, 2007.} \thanks{Partially supported by an NSF Postdoctoral Fellowship} \subjclass[2000]{53B20, 53A05, 35M10} \keywords{Local solvability; Monge-Amp\`{e}re equations; isometric embeddings} \begin{abstract} We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Amp\`{e}re type. These are: the problem of locally prescribed Gaussian curvature for surfaces in $\mathbb{R}^{3}$, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of degenerate Monge-Amp\`{e}re equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes and possesses a nonvanishing Hessian matrix at a critical point. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Let $K(u,v)$ be a function defined in a neighborhood of a point in $\mathbb{R}^{2}$, say $(u,v)=0$. A well-known problem is to ask, when does there exist a piece of a surface $z=z(u,v)$ in $\mathbb{R}^{3}$ having Gaussian curvature $K$? The classical results on this problem may be found in \cite{j1,p2,p3}. They show that a solution always exists when $K$ is analytic or $K$ does not vanish at the origin. In the case that $K\geq 0$ and is sufficiently smooth, or $K(0)=0$ and $|\nabla K(0)|\neq 0$, Lin provides an affirmative answer in \cite{l3,l4} (see \cite{h1} for a simplified proof of \cite{l4}). When $K\leq 0$ and $\nabla K$ possesses a certain nondegeneracy, Han, Hong, and Lin \cite{h5} show that a solution always exists. Furthermore, if $K$ degenerates to arbitrary finite order on a single smooth curve, then Han and the author independently provide an affirmative answer in \cite{h2,k1} (see also \cite{h3} for improved regularity). For an excellent survey of these results and related topics, see \cite{h4}. In this paper we prove the following, \begin{theorem} \label{thm1.1} Suppose that $K(0)=|\nabla K(0)|=0$, $\nabla^{2}K(0)$ has at least one negative eigenvalue, and $K\in C^{l}$, $l\geq 100$. Then there exists a piece of a $C^{l-98}$ surface in $\mathbb{R}^{3}$ with Gaussian curvature $K$. \end{theorem} If a surface in $\mathbb{R}^{3}$ is given by $z=z(u,v)$, then its Gaussian curvature is given by \begin{equation} \label{e1.1} z_{uu}z_{vv}-z^{2}_{uv}=K(1+|\nabla z|^{2})^{2}. \end{equation} Therefore our problem is equivalent to the local solvability of the above equation. Another well-known and related problem, is that of the local isometric embedding of surfaces into $\mathbb{R}^{3}$. That is, if $(M^{2},ds^{2})$ is a two-dimensional Riemannian manifold, when can one realize this, locally, as a small piece of a surface in $\mathbb{R}^{3}$? Suppose that $ds^{2}=Edu^{2}+2Fdudv+Gdv^{2}$ is given in the neighborhood of a point, say $(u,v)=0$. Then we must find three function $x(u,v)$, $y(u,v)$, $z(u,v)$, such that $ds^{2}=dx^{2}+dy^{2}+dz^{2}$. The following strategy was first used by Weingarten \cite{w1}. We search for a function $z(u,v)$, with $|\nabla z|$ sufficiently small, such that $ds^{2}-dz^{2}$ is flat in a neighborhood of the origin. Suppose that such a function exists, then since any Riemannian manifold of zero curvature is locally isometric to Euclidean space (via the exponential map), there exists a smooth change of coordinates $x(u,v)$, $y(u,v)$ such that $dx^{2}+dy^{2}=ds^{2}-dz^{2}$. Therefore, our problem is reduced to finding $z(u,v)$ such that $ds^{2}-dz^{2}$ is flat in a neighborhood of the origin. A computation shows that this is equivalent to the local solvability of the equation \begin{equation} \label{e1.2} (z_{11}-\Gamma_{11}^{i}z_{i})(z_{22}-\Gamma_{22}^{i}z_{i})- (z_{12}-\Gamma_{12}^{i}z_{i})^{2}=K(EG-F^{2}-Ez_{2}^{2}- Gz_{1}^{2}+2Fz_{1}z_{2}), \end{equation} where $z_{1}=\partial z/\partial u$, $z_{2}=\partial z/\partial v$, $z_{ij}$ are second partial derivatives of $z$, and $\Gamma_{jk}^{i}$ are Christoffel symbols. For this problem we obtain a similar result to that of Theorem \ref{thm1.1}. \begin{theorem} \label{thm1.2} Suppose that $K(0)=|\nabla K(0)|=0$, $\nabla^{2}K(0)$ has at least one negative eigenvalue, and $ds^{2}\in C^{l}$, $l\geq 102$. Then there exists a $C^{l-100}$ local isometric embedding into $\mathbb{R}^{3}$. \end{theorem} We note that Pogorelov has constructed a $C^{2,1}$ metric with no $C^{2}$ isometric embedding in $\mathbb{R}^{3}$. Other examples of metrics with low regularity not admitting a local isometric embedding have also been proposed by Nadirashvili and Yuan \cite{n1}. Furthermore, an alternate method for obtaining \textit{smooth} examples of local nonsolvability, for equations with similar structure, may be found in \cite{k2}. Equations \eqref{e1.1} and \eqref{e1.2} are both two-dimensional Monge-Amp\`{e}re equations. With the goal of treating both problems simultaneously, we will study the local solvability of the following general Monge-Amp\`{e}re equation \begin{equation} \label{e1.3} \det(z_{ij}+a_{ij}(u,v,z,\nabla z))=Kf(u,v,z,\nabla z), \end{equation} where $a_{ij}(u,v,p,q)$ and $f(u,v,p,q)$ are smooth functions of $p$ and $q$, $f>0$, and $a_{ij}(0,0,p,q)= \partial^{\alpha}a_{ij}(0,0,0,0)=0$, for any multi-index $\alpha$ in the variables $(u,v)$ satisfying $|\alpha|\leq 2$. Clearly \eqref{e1.1} is of the form \eqref{e1.3}, and \eqref{e1.2} is of the form \eqref{e1.3} if $\Gamma_{jk}^{i}(0)=0$, which we assume without loss of generality. We will prove \begin{theorem} \label{thm1.3} Suppose that $K(0)=|\nabla K(0)|=0$, $\nabla^{2}K(0)$ has at least one negative eigenvalue, and $K$, $a_{ij}$, $f\in C^{l}$, $l\geq 100$. Then there exists a $C^{l-98}$ local solution of \eqref{e1.3}. \end{theorem} \subsection*{Remark} (1) The methods carried out below may be slightly modified to yield the same result for the case when $\nabla^{2}K(0)$ has at least one positive eigenvalue; and therefore ultimately include the case of genuine second order vanishing, that is, when $K(0)=|\nabla K(0)|=0$ and $|\nabla^{2}K(0)|\neq 0$. It is conjectured that local solutions exist whenever $K$ vanishes to finite order and the $a_{ij}$ vanish to an order greater than or equal to half that of $K$. (2) Recently Han and the author \cite{h6} have shown that local solutions exist for the isometric embedding problem, whenever $K$ vanishes to finite order and the zero set $K^{-1}(0)$ consists of Lipschitz curves intersecting transversely at the origin. Unfortunately the methods of \cite{h6} breakdown when the transversality assumption is removed. Therefore Theorem \ref{thm1.3} (which allows tangential intersections) and the methods used to prove it, may be considered as a first step towards the general conjecture. Equation \eqref{e1.3} is elliptic if $K>0$, hyperbolic if $K<0$, and of mixed type if $K$ changes sign in a neighborhood of the origin. Furthermore, the order to which $K$ vanishes determines how \eqref{e1.3} changes type in the following way. If $K(0)=0$ and $|\nabla K(0)|\neq 0$ \cite{l4}, then \eqref{e1.3} is a nonlinear perturbation of the Tricomi equation: \[ vz_{uu}+z_{vv}=0. \] In our case, assuming that the origin is a critical point for which the Hessian matrix of $K$ does not vanish, \eqref{e1.3} is a nonlinear perturbation of Gallerstedt's equation \cite{g1}: \[ \pm v^{2}z_{uu}+z_{vv}=0. \] Therefore, if sufficiently small linear perturbation terms are added to the above two equations, then the first (second) partial $v$-derivative of the $z_{uu}$ coefficient will not vanish for the Tricomi (Gallerstedt) equation. It is this fact, which allows one to obtain appropriate estimates for the linearized equation of \eqref{e1.3} in both cases. This observation, Lemma \ref{lem2.3} below, is the key to our approach. From now on we only consider the case when $\nabla^{2}K(0)$ has at least one negative eigenvalue. Therefore, we can assume without loss of generality that \[ Kf(u,v,z,\nabla z)=-v^{2}+O(|u|^{2}+|v|^{3}+|z|^{2}+|\nabla z|^{2}). \] Let $\varepsilon$ be a small parameter and set $u=\varepsilon^{4}x$, $v=\varepsilon^{2}y$, $z=u^{2}/2-v^{4}/12+\varepsilon^{9}w$. Then substituting into \eqref{e1.3} and cancelling $\varepsilon^{5}$ on both sides yields \begin{equation} \label{e1.4} -y^{2}w_{xx}+w_{yy}+\varepsilon\widetilde{F}(\varepsilon,x,y,w, \nabla w,\nabla^{2} w)=0, \end{equation} where $\widetilde{F}(\varepsilon,x,y,p,q,r)$ is smooth with respect to $\varepsilon$, $p$, $q$, and $r$. Choose $x_{0}$, $y_{0}>0$ and define the rectangle $X=\{(x,y) : |x|< x_{0}, |y|< y_{0}\}$. Let $\psi \in C^{\infty}(X)$ be a cut-off function such that $$ \psi(x,y)=\begin{cases} 1 & \text{if $|x|\leq \frac{x_{0}}{2}$ and $|y|\leq \frac{y_{0}}{2}$},\\ 0 & \text{if $|x|\geq \frac{3x_{0}}{4}$ or $|y|\geq \frac{3y_{0}}{4}$}, \end{cases} $$ and cut-off the nonlinear term of (1.4) by $F(\varepsilon,x,y,w,\nabla w,\nabla^{2} w)=\psi\widetilde{F}$. Then solving \begin{equation} \label{e1.5} \Phi(w)=-y^{2}w_{xx}+w_{yy}+\varepsilon F(\varepsilon,x,y,w, \nabla w,\nabla^{2} w)=0\quad \text{in } X, \end{equation} is equivalent to solving \eqref{e1.3} locally at the origin. In the next sections, we shall study the linearization of \eqref{e1.5} about some function $w$. The linearized equation is a small perturbation of Gallerstedt's equation, which as mentioned above admits certain estimates. These estimates are sufficient for the existence of weak solutions, however the perturbation terms cause some difficulty in proving higher regularity. To avoid this problem, we will regularize the equation by appending a suitably small fourth order operator. In section $\S 2$ we shall prove the existence of weak solutions for a boundary value problem associated to this modified linearized equation. Regularity will be obtained in section $\S 3$. In section $\S 4$ we make the appropriate estimates in preparation for the Nash-Moser iteration procedure. Finally, in $\S 5$ we apply a modified version of the Nash-Moser procedure and obtain a solution of \eqref{e1.5}. \section{Linear Existence Theory} In this section we will prove the existence of weak solutions for a small perturbation of the linearized equation for \eqref{e1.5}. Fix a constant $\Lambda>0$, and for all $i,j=1,2$ let $b_{ij}$, $b_{i}$, $b\in C^{r}(\mathbb{R}^{2})$ be such that: \begin{itemize} \item[(i)] The supports of $b_{ij}$, $b_{i}$, and $b$ are contained in $X$, and \item[(ii)] $\sum |b_{ij}|_{C^{10}}+|b_{i}|_{C^{10}}+|b|_{C^{10}}\leq \Lambda$. \end{itemize} \noindent We will study the following generalization of the linearization for \eqref{e1.5}, \begin{equation} \label{e2.1} L=\sum_{i,j}a_{ij}\partial_{x_{i}x_{j}}+\sum_{i}a_{i}\partial_{x_{i}} +a \end{equation} where $x_{1}=x$, $x_{2}=y$ and $a_{11}=-y^{2}+\varepsilon b_{11}$, $a_{12}=\varepsilon b_{12}$, $a_{22}=1+\varepsilon b_{22}$, $a_{1}=\varepsilon b_{1}$, $a_{2}=\varepsilon b_{2}$, $a=\varepsilon b$. To simplify \eqref{e2.1}, we shall make a change of variables that will eliminate the mixed second derivative term. In constructing this change of variables we will make use of the following lemma from ordinary differential equations. \begin{lemma}[\cite{b1}] \label{lem2.1} Let $G(x,t)$ be a smooth real valued function in the closed rectangle $|x-s|\leq T_{1}$, $|t|\leq T_{2}$. Let $M=\sup|G(x,t)|$ in this domain. Then the initial-value problem $dx/dt=G(x,t)$, $x(0)=s$, has a unique smooth solution defined on the interval $|t|\leq \min(T_{2},T_{1}/M)$. \end{lemma} We now construct the desired change of variables. \begin{lemma} \label{lem2.2} For $\varepsilon$ sufficiently small, there exists a $C^{r}$ diffeomorphism \[ \xi=\xi(x,y), \quad \eta=y, \] of $X$ onto itself, such that in the new variables $(\xi,\eta)$ \[ L=\sum_{i,j}\overline{a}_{ij}\partial_{x_{i}x_{j}}+ \sum_{i}\overline{a}_{i}\partial_{x_{i}} +\overline{a}, \] where $x_{1}=\xi$, $x_{2}=\eta$, $\overline{a}_{11}=-\eta^{2}+\varepsilon \overline{b}_{11}$, $\overline{a}_{12}\equiv0$, $\overline{a}_{22}=1+\varepsilon \overline{b}_{22}$, $\overline{a}_{1}=\varepsilon \overline{b}_{1}$, $\overline{a}_{2}=\varepsilon \overline{b}_{2}$, $\overline{a}=\varepsilon\overline{b}$, and $\overline{b}_{ij}$, $\overline{b}_{i}$, $\overline{b}$ satisfy: \begin{itemize} \item[(i)] $\overline{b}_{ij},\overline{b}_{i},\overline{b} \in C^{r-2}(\overline{X})$, \item[(ii)] $\overline{b}_{ij}$, $\overline{b}_{i}$, and $\overline{b}$ vanish in a neighborhood of the lines $\xi=\pm x_{0}$, and \item[(iii)] $\sum |\overline{b}_{ij}|_{C^{8}(\overline{X})} +|\overline{b}_{i}|_{C^{8}(\overline{X})} +|\overline{b}|_{C^{8}(\overline{X})}\leq \Lambda'$, \end{itemize} for some fixed $\Lambda'$. \end{lemma} \begin{proof} Using the chain rule we find that $\overline{a}_{12}=a_{12}\xi_{x}+a_{22}\xi_{y}$. Therefore, we seek a smooth function $\xi(x,y)$ such that \begin{equation} \label{e2.2} a_{12}\xi_{x}+a_{22}\xi_{y}=0 \quad \text{in }X, \quad \xi(x,0)=x, \quad \xi(\pm x_{0},y)=\pm x_{0}. \end{equation} The boundary condition $\xi(\pm x_{0},y)=\pm x_{0}$ states that the vertical sides of $\partial X$ will be mapped identically onto themselves under the transformation $(\xi,\eta)$. Moreover, the horizontal portion of $\partial X$ will be mapped identically onto itself since $\eta=y$. Thus, $(\xi,\eta)$ will act as the identity map on $\partial X$. Since $a_{12}=\varepsilon b_{12}$ and $a_{22}=1+\varepsilon b_{22}$, by property (ii) if $\varepsilon$ is sufficiently small the line $y=0$ will be non-characteristic for \eqref{e2.2}. Then by the theory of first order partial differential equations, \eqref{e2.2} is reduced to the following system of first order ODE: \begin{gather*} \dot{x}=\frac{a_{12}}{a_{22}}, \quad x(0)=s, \quad -x_{0} \leq s\leq x_{0},\\ \dot{y}=1, \quad y(0)=0,\\ \dot{\xi}=0, \quad \xi(0)=s, \quad \xi(\pm x_{0},y)=\pm x_{0}, \end{gather*} where $x=x(t)$, $y=y(t)$, $\xi(t)=\xi(x(t),y(t))$ and $\dot{x}$, $\dot{y}$, $\dot{\xi}$ are derivatives with respect to $t$. We first show that the characteristic curves, given parametrically by $(x,y)=(x(t),t)$, exist globally for $-y_{0}\leq t\leq y_{0}$. We apply Lemma \ref{lem2.1} with $T_{1}=2x_{0}$ and $T_{2}=y_{0}$ to the initial-value problem $\dot{x}=\frac{a_{12}}{a_{22}}$, $x(0)=s$. By property (ii) for the $b_{ij}$ \[ M\leq \sup_{X}|\frac{a_{12}}{a_{22}}|=\varepsilon \sup_{X}|\frac{b_{12}}{1+\varepsilon b_{22}}|\leq \varepsilon C_{0}, \] so for $\varepsilon$ small, $M\leq \frac{2x_{0}}{y_{0}}$. Thus $\min(T_{2},T_{1}/M)=y_{0}$, and Lemma \ref{lem2.1} gives the desired global existence. We observe that $\xi=s$ is constant along each characteristic. In particular, since $\frac{a_{12}}{a_{22}}|_{(\pm x_{0},y)}=0$ the characteristics passing through $(\pm x_{0},0)$ are the vertical lines $(\pm x_{0},t)$, so that $\xi(\pm x_{0},y)=\pm x_{0}$ is satisfied. We now show that the map $\rho:X\to X$ given by \[ (s,t)\mapsto(x(s,t),y(s,t))=(x(s,t),t) \] is a diffeomorphism, from which we will conclude that $\xi=s(x,y)$ is a smooth function of $(x,y)$. To show that $\rho$ is 1-1, suppose that $\rho(s_{1},t_{1})=\rho(s_{2},t_{2})$. Then $t_{1}=t_{2}$ and $x(s_{1},t_{1})=x(s_{2},t_{2})$, which implies that $s_{1}=s_{2}$ by uniqueness for the initial-value problem for ordinary differential equations. To show that $\rho$ is onto, take an arbitrary point $(x_{1},y_{1})\in X$, then we will show that there exists $s\in [-x_{0},x_{0}]$ such that $\rho(s,y_{1})=(x(s,y_{1}),y_{1})=(x_{1},y_{1})$. Since the map $x(s,\cdot):[-x_{0},x_{0}]\to [-x_{0},x_{0}]$ is continuous and $x(\pm x_{0},\cdot)=\pm x_{0}$, the intermediate value theorem guarantees that there is $s\in [-x_{0},x_{0}]$ with $x(s,y_{1})=x_{1}$, showing that $\rho$ is onto. Therefore, $\rho$ has a well-defined inverse. To show that $\rho^{-1}$ is smooth it is sufficient, by the inverse function theorem, to show that the Jacobian of $\rho$ does not vanish at each point of $X$. Since \[ D\rho=\begin{pmatrix} x_{s} & x_{t} \\ 0 & 1 \\ \end{pmatrix}, \] this is equivalent to showing that $x_{s}$ does not vanish in $X$. Differentiate the equation for $x$ with respect to $s$ to obtain, $\frac{d}{dt}(x_{s})=(\frac{a_{12}}{a_{22}})_{x}x_{s}$, $x_{s}(0)=1$. Then by the mean value theorem, \[ |x_{s}(s,t)-1|=|x_{s}(s,t)-x_{s}(s,0)|\leq y_{0} \sup_{X}|(\frac{a_{12}}{a_{22}})_{x}|\sup_{X}|x_{s}| \] for all $(s,t)\in X$. Thus by property (ii) for the $b_{ij}$, \[ 1-\varepsilon C_{1} y_{0} \sup_{X}|x_{s}|\leq x_{s}(s,t)\leq \varepsilon C_{1} y_{0} \sup_{X}|x_{s}|+1 \] for all $(s,t)\in X$. Hence for $\varepsilon$ sufficiently small, $x_{s}(s,t)>0$ in $X$. We have now shown that $\rho$ is a diffeomorphism. Moreover, by Lemma \ref{lem2.1} and the inverse function theorem, we have $\rho,\rho^{-1}\in C^{r}$. Lastly we calculate $\overline{a}_{11}$, $\overline{a}_{22}$, $\overline{a}_{1}$, $\overline{a}_{2}$, and show that they possess the desired properties. It will first be necessary to estimate the derivatives of $\xi$. By differentiating \eqref{e2.2} with respect to $x$, we obtain \[ (\frac{a_{12}}{a_{22}})(\xi_{x})_{x}+(\xi_{x})_{y} =-(\frac{a_{12}}{a_{22}})_{x}\xi_{x}, \quad \xi_{x}(x,0)=1. \] As above, let $(x(t),y(t))$ be the parameterization of an arbitrary characteristic, then $\xi_{x}(t)=\xi_{x}(x(t),y(t))$ satisfies $\dot{\xi}_{x}=-(\frac{a_{12}}{a_{22}})_{x}\xi_{x}$, $\xi_{x}(0)=1$. By the mean value theorem, \[ |\xi_{x}(t)-1|=|\xi_{x}(t)-\xi_{x}(0)|\leq y_{0}\sup_{X}|(\frac{a_{12}}{a_{22}})_{x}| \sup_{X}|\xi_{x}|. \] By property (ii) for the $b_{ij}$, \[ 1-\varepsilon C_{1} y_{0} \sup_{X}|\xi_{x}|\leq \xi_{x}(t)\leq \varepsilon C_{1} y_{0} \sup_{X}|\xi_{x}|+1. \] Since this holds for any characteristic, we obtain \[ \sup_{X}|\xi_{x}|\leq \frac{1}{1-\varepsilon C_{1} y_{0}}:=C_{2}. \] It follows from \eqref{e2.2} that \[ \sup_{X}|\xi_{y}|\leq C_{3}, \] where $C_{2}$, $C_{3}$ are independent of $\varepsilon$ and $b_{ij}$. In order to estimate $\xi_{xx}$, differentiate \eqref{e2.2} two times with respect to $x$: \[ (\frac{a_{12}}{a_{22}})(\xi_{xx})_{x}+(\xi_{xx})_{y} =-2(\frac{a_{12}}{a_{22}})_{x}\xi_{xx}- (\frac{a_{12}}{a_{22}})_{xx}\xi_{x}, \quad \xi_{xx}(x,0)=0. \] Then the same procedure as above yields \[ \sup_{X}|\xi_{xx}|\leq \varepsilon C_{4} y_{0} \sup_{X}|\xi_{xx}|+\varepsilon C_{5} y_{0}, \] implying that \[ \sup_{X}|\xi_{xx}|\leq \frac{\varepsilon C_{5} y_{0}}{1-\varepsilon C_{4} y_{0}}:=\varepsilon C_{6}. \] Furthermore, using the above estimates we can differentiate \eqref{e2.2} to obtain \[ \sup_{X}|\xi_{xy}|\leq\varepsilon C_{7}, \quad \sup_{X}|\xi_{yy}|\leq\varepsilon C_{8}, \] for some constants $C_{7}$, $C_{8}$ independent of $\varepsilon$ and $b_{ij}$. This procedure may be continued to yield \[ |\partial^{\alpha}\xi|\leq \varepsilon C_{9}, \] for any multi-index $\alpha$ satisfying $2\leq|\alpha|\leq 10$. We now show that $\overline{a}_{11}$, $\overline{a}_{22}$, $\overline{a}_{1}$, $\overline{a}_{2}$ satisfy properties (i), (ii), (iii) and have the desired form. Calculation shows that, \[ \overline{a}_{11}=a_{11}\xi_{x}^{2}+2a_{12}\xi_{x}\xi_{y}+a_{22}\xi_{y}^{2}, \quad \overline{a}_{1}=a_{11}\xi_{xx}+2a_{12}\xi_{xy}+a_{22}\xi_{yy}+a_{1}\xi_{x} +a_{2}\xi_{y}. \] Furthermore, according to the above estimates and the fact that the $b_{ij}$ vanish in a neighborhood of $\partial X$, we may write \[ \xi_{x}=1+\varepsilon\chi, \] where $\chi\in C^{r-1}(\overline{X})$ vanishes in a neighborhood of the lines $x=\pm x_{0}$. It follows that \[ \overline{a}_{11}=-\eta^{2}+\varepsilon\overline{b}_{11}, \quad \overline{a}_{1}=\varepsilon\overline{b}_{1}, \] where $\overline{b}_{11}$ and $\overline{b}_{1}$ satisfy properties (i), (ii), (iii). Moreover since $\overline{a}_{22}=a_{22}$ and $\overline{a}_{2}=a_{2}$, properties (i), (ii), (iii) hold for these coefficients as well. \end{proof} For the remainder of this section and section $\S 3$, $(\xi,\eta)$ will be the coordinates of the plane. For simplicity of notation we put $x=\xi$, $y=\eta$, and $a_{ij}=\overline{a}_{ij}$, $a_{i}=\overline{a}_{i}$, $a=\overline{a}$, $b_{ij}=\overline{b}_{ij}$, $b_{i}=\overline{b}_{i}$, $b=\overline{b}$. To obtain a well-posed boundary value problem, we will study a regularization of $L$ in the infinite strip $\Omega=\{(x,y): |x|0$ is a small constant that will tend to zero in the Nash-Moser iteration procedure. Furthermore, we will need to modify some of the coefficients of $L$ away from $X$ as follows. First cut $b_{ij}$, $b_{i}$, and $b$ off near the lines $y=\pm y_{0}$, so that by property (ii) of Lemma \ref{lem2.2} these functions vanish in a neighborhood of $\partial X$, and the coefficients $a_{ij}$, $a_{i}$, and $a$ are now defined on all of $\Omega$. Choose values $y_{1}$, $y_{2}$, and $y_{3}$ such that $y_{0}0$ be a small constant that depends on $y_{2}-y_{1}$ and $y_{3}-y_{2}$. Then redefine the coefficient $a$ in the domain $\Omega-X$ so that: \begin{itemize} \item[(i)] $a\in C^{r-2}(\overline{\Omega})$, \item[(ii)] $a\equiv 1$ if $|y|\geq y_{1}$, \item[(iii)] $a\geq 0$ for $|y|\geq y_{0}$, \item[(iv)] $\partial_{y}a\geq 0$ if $y\geq y_{0}$, and $\partial_{y}a\leq 0$ if $y\leq -y_{0}$. \end{itemize} Redefine $a_{11}$ in $\Omega-X$ and near $\partial\Omega$ so that: \begin{itemize} \item[(i)] $a_{11}\in C^{r-2}(\overline{\Omega})$, \item[(ii)] $a_{11}=\begin{cases} -y^{2} & \text{if }y_{0}\leq |y|\leq y_{1},\\ -(\frac{y_{1}+y_{2}}{2})^{2} & \text{if }|y|\geq y_{2}, \end{cases}$ \item[(iii)] $\partial_{y}a_{11}<0$ if $y\geq y_{0}$, and $\partial_{y}a_{11}>0$ if $y\leq -y_{0}$, \item[(iv)] $\sup_{\Omega}\partial_{yy}a_{11}\leq \delta$, \item[(v)] $a_{11}|_{\partial\Omega}\leq-\theta$, $\partial_{x}^{\alpha}a_{11}|_{\partial\Omega}=0$, $\alpha\leq r-2$, and $\sup_{\Omega}|\partial_{x}^{\beta}a_{11}|\leq\varepsilon\Lambda'$, $1\leq\beta\leq 8$. \end{itemize} Lastly, redefine $a_{2}$ in $\Omega-X$ so that: \begin{itemize} \item[(i)] $a_{2}\in C^{r-2}(\overline{\Omega})$, \item[(ii)] $a_{2}=\begin{cases} 0 & \text{if }y_{0}\leq |y|\leq y_{2},\\ -\delta y+\delta(\frac{y_{2}+y_{3}}{2}) & \text{if }y\geq y_{3},\\ -\delta y-\delta(\frac{y_{2}+y_{3}}{2}) & \text{if }y\leq -y_{3}, \end{cases}$ \item[(iii)] $a_{2}\leq 0$ if $y\geq y_{2}$, and $a_{2}\geq 0$ if $y\leq -y_{2}$, \item[(iv)] $\sup_{|y|\geq y_{2}}|\partial_{y}a_{2}|\leq\delta$. \end{itemize} Denote the operator $L$ with coefficients modified as above by $L'$, and define \[ L_{\theta}=-\theta\partial_{xxyy}+L'. \] Note that since we are studying a local problem, as stated in the introduction, we may modify the coefficients of the linearization away from a fixed neighborhood of the origin. This will become clear in the final section, where a modified version of the Nash-Moser iteration scheme is used. Consider the following boundary value problems \begin{gather} L_{\theta}u=f \quad \text{in } \Omega, \quad u|_{\partial\Omega}=0; \label{e2.3}\\ L_{\theta}u=f \quad \text{in } \Omega, \quad u_{x}|_{\partial\Omega}=0, \label{e2.4} \end{gather} and the corresponding adjoint problems \begin{gather} L_{\theta}^{*}v=g \quad \text{in } \Omega, \quad v|_{\partial\Omega}=0; \label{e2.5}\\ L_{\theta}^{*}v=g \quad \text{in }\Omega, \quad v_{x}|_{\partial\Omega}=0, \label{e2.6} \end{gather} where $L_{\theta}^{*}$ is the formal adjoint of $L_{\theta}$. The main result of this section is to obtain weak solutions for all four problems. We will make extensive use of the following function spaces. For $m,n\in \mathbb{Z}_{\geq 0}$ let \begin{align*} C^{(m,n)}(\overline{\Omega}) &=\{u:\Omega\to \mathbb{R}: \partial_{x}^{\alpha}\partial_{y}^{\beta}u\in C^{0}(\overline{\Omega}),\text{ $\alpha\leq m$, $\beta\leq n$}\},\\ \widetilde{C}^{(m,n)}(\overline{\Omega}) &=\{u\in C^{(m,n)}(\overline{\Omega}): u|_{\partial\Omega}=0,\text{ $u$ has bounded support}\},\\ \widetilde{C}_{x}^{(m,n)}(\overline{\Omega}) &= \{u\in C^{(m,n)}(\overline{\Omega}): u_{x}|_{\partial\Omega}=0,\text{ $u$ has bounded support}\}. \end{align*} Define the norm $$ \|u\|_{(m,n)}=\Big(\sum_{\alpha\leq m, \beta\leq n}\| \partial_{x}^{\alpha}\partial_{y}^{\beta}u\|_{L^{2}(\Omega)}^{2}\Big)^{1/2}, $$ and let $\widetilde{H}^{(m,n)}(\Omega)$ and $\widetilde{H}_{x}^{(m,n)}(\Omega)$ be the respective closures of $\widetilde{C}^{(m,n)}(\overline{\Omega})$ and $\widetilde{C}_{x}^{(m,n)}(\overline{\Omega})$ in the norm $\|\cdot\|_{(m,n)}$. Furthermore, let $H^{m}(\Omega)$ denote the Sobolev space of square integrable derivatives up to and including order $m$, with norm $\|\cdot\|_{m}$. Denote the $L^{2}(\Omega)$ inner product and norm by $(\cdot,\cdot)$ and $\|\cdot\|$ respectively, and define the negative norm \[ \| u\|_{(-m,-n)}=\sup_{v\in \widetilde{H}^{(m,n)}(\Omega)}\frac{|(u,v)|}{\|v\|_{(m,n)}}. \] Let $\widetilde{H}^{(-m,-n)}(\Omega)$ be the closure of $L^{2}(\Omega)$ in the norm $\|\cdot\|_{(-m,-n)}$, then $\widetilde{H}^{(-m,-n)}(\Omega)$ is the dual space of $\widetilde{H}^{(m,n)}(\Omega)$. The dual space of $\widetilde{H}_{x}^{(m,n)}(\Omega)$ is defined similarly. Let $f\in L^{2}(\Omega)$. A function $u\in L^{2}(\Omega)$ is said to be a weak solution of \eqref{e2.3} (respectively \eqref{e2.4}) if \[ (u,L_\theta^{*}v)=(f,v), \text{ for }\text{ all } v\in \widetilde{C}^{\infty}(\overline{\Omega}) \text{ (for }\text{ all }v\in \widetilde{C}_{x}^{\infty}(\overline{\Omega})). \] We shall employ the energy integral method, developed by K. O. Friedrichs and others, to prove the existence of weak solutions for \eqref{e2.3} and \eqref{e2.4}. The first step is to establish an a priori estimate. \begin{lemma}[Basic Estimate] \label{lem2.3} If $\varepsilon$, $\theta$, and $\delta$ are sufficiently small, then there exist constants $C_{1},C_{2}>0$ independent of $\varepsilon$, $\theta$, $\delta$, and functions $A,B,C,D,E\in C^{\infty}(\overline{\Omega})$ where $E>0$ and $E=O(|y|)$ as $|y|\to\infty$, such that: \begin{align*} &(Au+Bu_{x}+Cu_{y}+Du_{yy},L_{\theta}u)\\ &\geq C_{1}[\| u\|^{2}+\| E u_{y}\|^{2}+\theta(\| u_{x}\|^{2}+\| u_{xy}\|^{2}+\| u_{yy}\|^{2}+\theta\| u_{xyy}\|^{2})], \end{align*} for all $u\in C^{\infty}(\overline{\Omega})$ with bounded support such that $u_{x}(-x_{0},y)=0$, and either $u(x_{0},y)=0$ or $u_{x}(x_{0},y)=0$. Furthermore, \[ \| u\|+\| u_{y}\|+\sqrt{\theta}(\| u_{x}\|+\| u_{xy}\|+\| u_{yy}\|+\sqrt{\theta}\| u_{xyy}\|)\leq C_{2}\| L_{\theta}u\|, \] for all $u\in \widetilde{C}^{\infty}(\overline{\Omega})$ and for all $u\in \widetilde{C}^{\infty}_{x}(\overline{\Omega})$. \end{lemma} \begin{proof} We first define the functions $A,B,C$ and $D$. Let $\mu$ be a positive constant such that $\frac{1}{4}\mu+ a_{11}\geq 1$ throughout $\Omega$, and let $\gamma\in C^{\infty}([-x_{0},x_{0}])$ be such that $$ \gamma(x)= \begin{cases} 1 & \text{if $-x_{0}\leq x\leq\frac{x_{0}}{2}$},\\ 0 & \text{if $x=x_{0}$}, \end{cases} $$ with $\gamma(x)>0$ except at $x=x_{0}$, and $\gamma'\leq 0$. Define \begin{gather*} A=\frac{1}{2}\partial_{y}C-a_{11}, \quad B=-\theta\gamma, \\ C=\begin{cases} \mu\partial_{y}a_{11} & \text{if $|y|0$ independent of $\varepsilon$ and $\theta$. In order to accomplish this we shall treat the regions $|y|\leq y_{0}$, $y_{0}\leq |y|\leq y_{1}$, $y_{1}\leq |y|\leq y_{2}$, and $|y|\geq y_{2}$ separately. Moreover throughout this proof $M_{i}$, $i=1,2,\dots$, will always denote positive constants independent of $\varepsilon$ and $\theta$. A computation yields, \[ I_{10}=-a_{22}\partial_{yy}a_{11}- a_{11}a-\frac{1}{2}C\partial_{y}a-\frac{1}{2}(Aa_{2})_{y} +O(\varepsilon+\theta). \] In the region $|y|\leq y_{0}$ we have $a,\partial_{y}a,a_{2},\partial_{y}a_{2}=O(\varepsilon)$, $a_{22}=1+O(\varepsilon)$, and $\partial_{yy} a_{11}=-2+O(\varepsilon)$, so that here $I_{10}\geq M_{2}$. If $y_{0}\leq |y|\leq y_{1}$, the conditions placed on $a$ guarantee that \[ -a_{11}a-\frac{1}{2}C\partial_{y}a\geq 0; \] furthermore $a_{22}$, $a_{11}$, and $a_{2}$ have the same properties in this region as in the previous. Hence, $I_{10}\geq M_{3}$ when $y_{0}\leq |y|\leq y_{1}$. If $y_{1}\leq |y|\leq y_{2}$ then \[ -a_{22}\partial_{yy}a_{11}=O(\delta), \quad -a_{11}a\geq y_{1}^{2}, \quad a_{2}=\partial_{y}a\equiv 0, \] showing that $I_{10}\geq M_{4}$ in this region. Lastly, when $|y|\geq y_{2}$ we have $I_{10}\geq M_{5}$ since \[ \partial_{yy}a_{11}=\partial_{y}a\equiv 0, \quad -a_{11}a=(\frac{y_{1}+y_{2}}{2})^{2}, \quad -\frac{1}{2}(Aa_{2})_{y}=O(\delta). \] The desired conclusion now follows by combining the above estimates. Next we show that \[ \iint_{\Omega}I_{2}u_{yy}^{2}+2I_{3}u_{yy}u_{xy}+I_{4}u_{xy}^{2}\geq M_{6}\theta(\| u_{yy}\|^{2}+\| u_{xy}\|^{2}), \] where \[ I_{2}=-\frac{1}{2}\theta D_{xx}+Da_{22}, \quad I_{3}=-\frac{1}{2}\theta C_{x}, \quad I_{4}=-\frac{1}{2}\theta C_{y}-\frac{1}{2}\theta B_{x}-\theta A+Da_{11}. \] This will follow if $I_{2}\geq M_{7}\theta$, $I_{4}\geq M_{8}\theta$, and $I_{2}I_{4}-I_{3}^{2}>0$. A calculation shows that \begin{gather*} I_{2}=\theta a_{22}=\theta(1+O(\varepsilon)), \quad I_{3}=O(\varepsilon\theta), \\ I_{4}=2\theta( a_{11}-\frac{1}{2}C_{y})+O(\varepsilon\theta)=2\theta(\mu+ a_{11}+O(\varepsilon)). \end{gather*} Therefore since $\mu$ was chosen so that $\mu+a_{11}\geq 1$ in $\Omega$, the desired conclusion follows if $\varepsilon$ is sufficiently small. We now show that \[ \iint_{\Omega}I_{7}u_{x}^{2}+2I_{8}u_{x}u_{y}+I_{9}u_{y}^{2}\geq M_{9}(\theta\| u_{x}\|^{2}+\| E u_{y}\|^{2}), \] where \begin{gather*} 2I_{7}=-2Aa_{11}-(Ba_{11})_{x}+2Ba_{1}+(Ca_{11})_{y}+\theta B_{xyy}+\theta A_{yy}-(Da_{11})_{yy}, \\ 2I_{8}=-(Ba_{22})_{y}+Ba_{2}-(Ca_{11})_{x} +Ca_{1}+\theta A_{xy}+(Da_{11})_{xy}-(Da_{1})_{y}, \\ \begin{aligned} 2I_{9}&=-2Aa_{22}-(Ca_{22})_{y}+ 2Ca_{2}+\theta C_{xxy}+\theta A_{xx}\\ &\quad -(Da_{11})_{xx}-(Da_{2})_{y}+(Da_{1})_{x}-2Da. \end{aligned} \end{gather*} Again this will follow if $I_{7}\geq M_{10}\theta$, $I_{9}\geq M_{11}E^{2}$, and $I_{7}I_{9}-I_{8}^{2}>0$. A calculation shows that \begin{gather*} I_{7}=a_{11}^{2}+\frac{1}{2}C\partial_{y}a_{11} +\theta(-\partial_{yy}a_{11}+\frac{1}{2}\gamma_{x}a_{11}+O(\varepsilon)), \\ I_{8}=-\frac{1}{2}C_{x}a_{11}-\frac{1}{2}C\partial_{x}a_{11} +\frac{1}{2}Ca_{1}+\frac{1}{2}Ba_{2}+O(\theta), \\ \begin{aligned} I_{9}&=(a_{11}-C_{y})a_{22}+Ca_{2}+O(\varepsilon+\theta)\\ &=(2\mu+a_{11}+O(\varepsilon))(1+O(\varepsilon))+Ca_{2} +O(\varepsilon+\theta). \end{aligned} \end{gather*} Then $I_{9}\geq M_{11}E^{2}$ immediately follows since $Ca_{2}=O(\varepsilon)$ if $|y|\leq y_{0}$, $Ca_{2}\geq 0$ if $|y|\geq y_{0}$, $Ca_{2}=O(|y|^{2})$ as $|y|\to\infty$, and $2\mu+a_{11}\geq 1$. To show that $I_{7}\geq M_{10}\theta$, we consider the regions $|y|\leq y_{0}$ and $|y|\geq y_{0}$ separately. If $|y|\leq y_{0}$ then \[ C\partial_{y}a_{11}=\mu(\partial_{y}a_{11})^{2}\geq 0, \quad -\partial_{yy}a_{11}=2+O(\varepsilon), \quad \gamma_{x}a_{11}\geq-O(\varepsilon), \] so that here $I_{7}\geq 2\theta+O(\varepsilon\theta)$. Furthermore, when $|y|\geq y_{0}$ we have $I_{7}\geq y_{0}^{4}+O(\theta)$ since \[ a_{11}^{2}\geq y_{0}^{4}, \quad C\partial_{y}a_{11}\geq 0. \] Finally, $I_{7}I_{9}-I_{8}^{2}>0$ follows from the next calculation. If $|y|\leq y_{0}$ then \begin{align*} I_{7}I_{9}-I_{8}^{2} &\geq(a_{11}^{2} +\frac{\mu}{2}(\partial_{y}a_{11})^{2}+2\theta +O(\varepsilon\theta))(1+O(\varepsilon+\theta)) \\ &\quad -\frac{1}{4}O(\varepsilon^{2})a_{11}^{2} -\frac{1}{4}O(\varepsilon^{2})(\partial_{y} a_{11})^{2}-O(\varepsilon\theta+\theta^{2}), \end{align*} whereas if $|y|\geq y_{0}$ then \[ I_{7}I_{9}-I_{8}^{2}\geq(y_{0}^{4} +O(\theta))(1+O(\delta y^{2}))-O(\theta^{2}y^{2}). \] Lastly we deal with the term $2I_{6}u_{xy}u_{y}$. Consider the quadratic form: \[ M_{6}\theta u_{xy}^{2}+2I_{6}u_{xy}u_{y}+M_{9}E^{2}u_{y}^{2}, \] where $I_{6}=-\frac{1}{2}Ba_{22}$. Since \[ (M_{2}\theta)(M_{3}E^{2})-I_{6}^{2}\geq M_{11}\theta-M_{12}\theta^{2} (1+O(\varepsilon)) \] for some $M_{11}$, $M_{12}$, we obtain \[ M_{6}\theta u_{xy}^{2}+2I_{6}u_{xy}u_{y}+M_{9}E^{2}u_{y}^{2}\geq M_{13}(\theta u_{xy}^{2}+E^{2}u_{y}^{2}). \] This completes the proof of the first estimate. To obtain the second estimate we need only observe that the above arguments hold if $B\equiv 0$ and $u\in\widetilde{C}^{\infty}(\overline{\Omega})$ or $u\in\widetilde{C}^{\infty}_{x}(\overline{\Omega})$. Then an application of Cauchy's inequality ($ab\leq\lambda a^{2}+\frac{1}{4\lambda}b^{2}$, $\lambda>0$) yields the desired result. The reason for including $B$ in the first estimate will soon become clear. \end{proof} Having established the basic estimate, our goal shall now be to establish dual inequalities of the form: \begin{gather*} \| v\|\leq C_{1}\| L_{\theta}^{*}v\|_{(-1,-2)} \quad \text{for all } v\in \widetilde{C}^{\infty}(\overline{\Omega}), \\ \| v\|\leq C_{2}\| L_{\theta}^{*}v\|_{(-1,-2)} \quad \text{for all } v\in \widetilde{C}_{x}^{\infty}(\overline{\Omega}). \end{gather*} The existence of weak solutions to problems \eqref{e2.3} and \eqref{e2.4} will then easily follow from these two dual estimates, respectively. In order to establish the dual estimates, we will need the following lemma. Let $P$ denote the differential operator \[ P=D\partial_{y}^{2}+B\partial_{x}+C\partial_{y}+A, \] where $A,B,C,$ and $D$ are defined in Lemma \ref{lem2.3}. Note that $P$ is parabolic in $\Omega$, away from the portion of the boundary, $x=x_{0}$. This is the reason for including $B$ in the first estimate of Lemma \ref{lem2.3}. \begin{lemma} \label{lem2.4} For every $v\in \widetilde{C}^{\infty}(\overline{\Omega})$ there exists a unique solution $u\in C^{\infty}(\Omega)\cap H^{4}(\Omega)\subset C^{\infty}(\Omega)\cap C^{2}(\overline{\Omega})$ of \[ Pu=v \quad\textit{in } \Omega, \quad u(-x_{0},y)=u_{x}(-x_{0},y)=0, \quad u(x_{0},y)=0. \] Furthermore, for every $v\in \widetilde{C}_{x}^{\infty}(\overline{\Omega})$ there exists a unique solution $u\in C^{\infty}(\Omega)\cap H^{4}(\Omega)\subset C^{\infty}(\Omega)\cap C^{2}(\overline{\Omega})$ of \[ Pu=v \quad \textit{in } \Omega, \quad u_{x}(-x_{0},y)=0, \quad u_{x}(x_{0},y)=0. \] \end{lemma} \begin{proof} Let $\tau>0$ be a small parameter, and define the subdomains \[ \Omega_{\tau}=\{(x,y): -x_{0}0$. Also $\kappa D>0$, \[ -D\zeta+\kappa(\frac{1}{2}B_{x} -\frac{1}{2}C_{y}-A)\geq \kappa(2\mu+a_{11}+O(\varepsilon+\theta))\geq \kappa, \] and \begin{align*} &\frac{1}{2}\kappa A_{yy}+\frac{1}{2}(D\zeta)_{yy}- \frac{1}{2}(B\zeta)_{x}-\frac{1}{2}(C\zeta)_{y}+\zeta A \\ &=-\frac{1}{2}\kappa\partial_{yy}a_{11}-\frac{1}{2}C\zeta_{y} -\zeta a_{11}+\frac{1}{2}(D\zeta)_{yy}-\frac{1}{2} (B\zeta)_{x}+O(\varepsilon) \\ &\geq \begin{cases} \kappa-\zeta a_{11}+O(\varepsilon+\theta) & \text{if $|y|\leq y_{1}$},\\ |y|^{-1/2}[\frac{1}{2}\mu+a_{11}+O(\theta)]+O(\kappa\delta) & \text{if $y_{1}\leq |y|\leq y_{2}$,}\\ |y|^{-1/2}[\frac{1}{2}\mu+a_{11}+O(\theta)] & \text{if $|y|\geq y_{2}$.} \end{cases} \end{align*} Therefore if $\varepsilon$, $\theta$, and $\delta$ are sufficiently small, we may apply the Schwarz inequality followed by Cauchy's inequality to obtain \[ \|\sqrt{-\zeta}u^{k}\|_{\Omega_{\tau}}+\| u^{k}_{y}\|_{\Omega_{\tau}}+\| u^{k}_{yy}\|_{\Omega_{\tau}}\leq M_{1}\| Pu^{k}\|_{\Omega_{\tau}}, \] for some constant $M_{1}$ independent of $\tau$. The properties of $u^{k}$ guarantee that by letting $k\to\infty$, we obtain \[ \|\sqrt{-\zeta}u\|_{\Omega_{\tau}}+\| u_{y}\|_{\Omega_{\tau}}+\| u_{yy}\|_{\Omega_{\tau}}\leq M_{1}\| Pu\|_{\Omega_{\tau}}=M_{1}\| v\|_{\Omega_{\tau}}\leq M_{1}\| v\|. \] We now estimate $\partial_{x}^{\alpha}\partial_{y}^{\beta}u$ for $\alpha=1,\dots,4$, and $\beta=0,1,2$. Differentiate $Pu=v$ with respect to $x$: \begin{equation} \label{e2.8} D(u_{x})_{yy}+B(u_{x})_{x}+C(u_{x})_{y}+(A+B_{x})u_{x} =v_{x}-C_{x}u_{y}-A_{x}u. \end{equation} Since $u_{x}(-x_{0},y)=0$ and $A_{x}$, $C_{x}$ vanish outside a compact set, we can apply the same procedure as above to obtain \begin{align*} \|\sqrt{-\zeta}u_{x}\|_{\Omega_{\tau}}+\| u_{xy}\|_{\Omega_{\tau}}+\| u_{xyy}\|_{\Omega_{\tau}} &\leq M_{1}\| v_{x}-C_{x}u_{y}-A_{x}u\|_{\Omega_{\tau}}\\ &\leq M_{2}(\| v_{x}\|_{\Omega_{\tau}}+ \| u_{y}\|_{\Omega_{\tau}}+\| u \|_{\Omega_{\tau}})\\ &\leq M_{3}(\| v\|+\| v_{x}\|). \end{align*} Differentiating \eqref{e2.8} with respect to $x$ produces \begin{align*} &D(u_{xx})_{yy}+B(u_{xx})_{x}+C(u_{xx})_{y}+(A+2B_{x})u_{xx} \\ &=v_{xx}-\partial_{x}(C_{x}u_{y}+A_{x}u) -C_{x}u_{xy}-(A_{x}+B_{xx})u_{x}:=v_{1}. \end{align*} Again we apply the same method. However since $u_{xx}(-x_{0},y) =B^{-1}v_{x}|_{(-x_{0},y)}$ from \eqref{e2.8}, we now have \begin{align*} \|\sqrt{-\zeta}u_{xx}\|_{\Omega_{\tau}} + \| u_{xxy}\|_{\Omega_{\tau}} + \| u_{xxyy}\|_{\Omega_{\tau}} &\leq M_{1}\| v_{1}\|_{\Omega_{\tau}} +M_{4}\\ &\leq M_{5}(\| v\| + \| v_{x}\| + \| v_{xx}\|) + M_{4}, \end{align*} where $M_{4}=\kappa|B|^{-1}(\int_{x=-x_{0}}v_{xy}^{2} +v_{x}^{2})^{1/2}$ which is independent of $\tau$. We can estimate $\|\sqrt{-\zeta}\partial_{x}^{\alpha}u\|_{\Omega_{\tau}}$, $\alpha=3,4$, and $\|\partial_{x}^{\alpha}\partial_{y}^{\beta}u\|_{\Omega_{\tau}}$, $\alpha=3,4$, $\beta=1,2$, in a similar manner. To estimate $u_{yyy}$, differentiate $Pu=v$ with respect to $y$: \begin{equation} \label{e2.9} D(u_{y})_{yy}+B(u_{y})_{x}+C(u_{y})_{y}+(A+C_{y})u_{y} =v_{y}-A_{y}u. \end{equation} Since $u_{y}(-x_{0},y)=0$, $C_{y}<0$, and $A_{y}$ vanishes outside a compact set, the same method as above yields \begin{align*} \|\sqrt{-\zeta}u_{y}\|_{\Omega_{\tau}}+\| u_{yy}\|_{\Omega_{\tau}}+\| u_{yyy}\|_{\Omega_{\tau}} &\leq M_{1}\| v_{y}-A_{y}u\|_{\Omega_{\tau}}\\ &\leq M_{6}(\| v\|+\| v_{y}\|). \end{align*} Furthermore, $\| u_{xyyy}\|_{\Omega_{\tau}}$ and $\| u_{yyyy}\|_{\Omega_{\tau}}$ can be estimated by differentiating \eqref{e2.9} with respect to $x$ and $y$, respectively. The combination of all the above estimates produces, \[ \sum_{\alpha=0}^{4}\|\sqrt{-\zeta}\partial_{x}^{\alpha}u\|_{\Omega_{\tau}} +\sum_{\alpha+\beta\leq 4,\; \beta\neq 0} \|\partial_{x}^{\alpha}\partial_{y}^{\beta}u\|_{\Omega_{\tau}} \leq M_{7}\| v\|_{4}+M_{8}, \] where $M_{7}$ and $M_{8}$ are independent of $\tau$. Then letting $\tau \to 0$ we find that $\partial_{x}^{\alpha}\partial_{y}^{\beta}u\in L^{2}(\Omega)$, $\alpha+\beta\leq 4$, $\beta\neq 0$, and that $\sqrt{-\zeta}\partial_{x}^{\alpha}u\in L^{2}(\Omega)$, $\alpha=0,\dots,4$. It follows that $u\in H^{4}(K)$ for every compact $K\subset\Omega$, so that $u\in C^{2}(\overline{\Omega})$. We now show that $\partial_{x}^{\alpha}u\in L^{2}(\Omega)$, $\alpha=0,\dots,4$. Let $\varrho_{1},\varrho_{2}\in C^{\infty}(\mathbb{R})$ be given by $$ \varrho_{1}(x)= \begin{cases} -B+\theta & \text{if $-x_{0}\leq x\leq\frac{-x_{0}}{2}$},\\ 0 & \text{if $0\leq x\leq x_{0}$.} \end{cases} \quad \varrho_{2}(y)= \begin{cases} -y & \text{if $|y|\leq y_{0}$,}\\ 0 & \text{if $|y|\geq T$,} \end{cases} $$ such that $\varrho_{2}(y)\leq 0$ if $y>0$ and $\varrho_{2}(y)\geq 0$ if $y<0$, where $T>0$ is large enough so that $-1\leq\varrho_{2}'\leq\varepsilon$. Then define $\overline{B}=B+\varrho_{1}$ and $\overline{C}=C+\varrho_{2}-\varepsilon\mu\partial_{y} b_{11}=-2\mu y+\varrho_{2}$, and set \[ \overline{P}=\overline{B}\partial_{x}+\overline{C}\partial_{y}+A. \] If $w\in C_{c}^{\infty}(\overline{\Omega})$, then integrating by parts yields \[ (w,\overline{P}^{*}w)=\iint_{\Omega}[-\frac{1}{2} \overline{B}_{x}-\frac{1}{2}\overline{C}_{y}+A]w^{2} +\int_{\partial\Omega}[-\frac{1}{2}\overline{B}n_{1}]w^{2}. \] The boundary integral is nonnegative since $\overline{B}(-x_{0},y)=\theta$ and $\overline{B}(x_{0},y)=0$. Furthermore \[ -\frac{1}{2} \overline{B}_{x}-\frac{1}{2}\overline{C}_{y}+A =-\varrho_{2}'-a_{11}+O(\varepsilon+\theta)\geq M_{9}, \] for some constant $M_{9}>0$. Thus \begin{equation} \label{e2.10} \| w\|\leq M_{10}\|\overline{P}^{*}w\|. \end{equation} Since $v-Du_{yy}+\varrho_{1}u_{x}+(\varrho_{2}-\varepsilon\mu\partial_{y} b_{11})u_{y}\in L^{2}(\Omega)$, \eqref{e2.10} implies (see the proof of Theorem \ref{thm2.1} below) the existence of a weak solution $\widetilde{u}\in L^{2} (\Omega)$ of \[ \overline{P}\widetilde{u}=v-Du_{yy}+\varrho_{1}u_{x}+(\varrho_{2}-\varepsilon\mu\partial_{y} b_{11})u_{y}, \quad \widetilde{u}(-x_{0},y)=0. \] We shall now show that $u\equiv\widetilde{u}$. Since $\overline{P}$ is a first order differential operator, we may apply Peyser's extension \cite{p4} of Friedrichs' result \cite{f1} on the identity of weak and strong solutions to obtain a sequence $\{\widetilde{u}^{k}\}_{k=1}^{\infty}$, such that $\widetilde{u}^{k}\in C^{\infty}(\overline{\Omega})$ has bounded support, satisfies $\widetilde{u}^{k}(-x_{0},y)=0$, and \[ \|\widetilde{u}-\widetilde{u}^{k}\|+ \|\overline{P}\widetilde{u}^{k}-(v-Du_{yy}+\varrho_{1}u_{x}+(\varrho_{2}-\varepsilon \mu\partial_{y}b_{11})u_{y})\|\to 0 \text{ as } k\to\infty. \] Set $v^{k}=u-\widetilde{u}^{k}$. Using the fact that $|y|^{-1/4}v^{k}\to|y|^{-1/4}(u-\widetilde{u})\in L^{2}(\Omega)$ and recalling the definition of $\overline{P}$, we have \begin{align*} |(-|y|^{-1/4}v^{k},\overline{P}v^{k})| &\leq \||y|^{-1/4}v^{k}\|\|\overline{P}v^{k}\|\\ &\leq M_{11}\| v-Du_{yy}+\varrho_{1}u_{x}+(\varrho_{2} -\varepsilon\mu\partial_{y} b_{11})u_{y}-\overline{P}\widetilde{u}^{k}\|\to 0. \end{align*} Then the following calculation shows that $\| u-\widetilde{u}^{k}\|_{L^{2}(K)}\to 0$ for every compact $K\subset\Omega$: \begin{align*} (-|y|^{-1/4}v^{k},\overline{P}v^{k}) &= \lim_{t\to\infty} \iint_{\Omega^{(0,t)}}[\frac{1}{2}|y|^{-1/4}\overline{B}_{x}+ \frac{1}{2}(|y|^{-1/4}\overline{C})_{y}-|y|^{-1/4}A](v^{k})^{2}\\ &\quad + \int_{\partial\Omega^{(0,t)}}[-\frac{1}{2}|y|^{-1/4}\overline{C}n_{2} -\frac{1}{2}|y|^{-1/4}\overline{B}n_{1}](v^{k})^{2}\\ &\geq \lim_{t\to\infty} \iint_{\Omega^{(0,t)}}[|y|^{-1/4}(\frac{1}{4}\mu+ a_{11}-\frac{1}{2}+O(\varepsilon+\theta))](v^{k})^{2}\\ &\geq M_{12}\||y|^{-1/8}v^{k}\|_{K}^{2}. \end{align*} Therefore, $u\equiv\widetilde{u}$ in $L^{2}(\Omega)$. Differentiating the equation $Pu=v$ with respect to $\partial_{x}^{\alpha}$, $\alpha=1,\dots,4$, and applying the above procedure shows that $\partial_{x}^{\alpha}u\in L^{2}(\Omega)$, $\alpha=1,\dots,4$. We now have that $u\in H^{4}(\Omega)$. To complete the case when $v\in \widetilde{C}^{\infty}(\overline{\Omega})$, we must show that $u(x_{0},y)=0$. Since $B(x_{0},y)=0$, from the equation $Pu=v$ we find that \[ (Du_{yy}+Cu_{y}+Au)|_{(x_{0},y)}=v(x_{0},y)=0. \] Furthermore since $u\in H^{4}(\Omega)$, $u\to 0$ as $|y|\to\infty$. Therefore by applying the maximum principle to the above equation, we have $u(x_{0},y)=0$. We now consider the case when $v\in\widetilde{C}_{x} ^{\infty}(\overline{\Omega})$. Let $h(y)\in H^{\infty}(\mathbb{R})$ be the unique solution of the ODE: \[ D(-x_{0},y)h''+C(-x_{0},y)h'+A(-x_{0},y)h=v(-x_{0},y). \] Then as before, the parabolicity of $P$ guarantees the existence of a unique solution to the Cauchy problem \[ Pu=v \quad \text{in } \Omega, \quad u(-x_{0},y)=h(y), \] such that $u\in H^{\infty}(\Omega_{\tau})$ for every $\tau$. Furthermore, $u_{x}(-x_{0},y)=0$ since \[ Bu_{x}|_{(-x_{0},y)}=v(-x_{0},y)-(Du_{yy}+Cu_{y}+Au)|_{(-x_{0},y)}=0. \] Moreover, the same methods used above can be used here to show that $u\in H^{4}(\Omega)$. Lastly to show that $u_{x}(x_{0},y)=0$, differentiate $Pu=v$ with respect to $x$ and use that $B(x_{0},y)=0$ to obtain \[ (D(u_{x})_{yy}+C(u_{x})_{y}+(A+B_{x})u_{x})|_{(x_{0},y)}=v_{x}(x_{0},y) -(C_{x}u_{y}+A_{x}u)|_{(x_{0},y)}=0. \] Since $u_{x}\to 0$ as $|y|\to\infty$, by the maximum principle $u_{x}(x_{0},y)=0$.\end{proof} With Lemma \ref{lem2.4} we are now in a position to establish the dual inequalities. \begin{proposition} \label{prop2.1} There exist constants $M_{1}$, $M_{2}$ such that: \begin{gather*} \| v\|\leq M_{1}\| L_{\theta}^{*}v\|_{(-1,-2)} \textit{ for all } v\in \widetilde{C}^{\infty}(\overline{\Omega}), \\ \| v\|\leq M_{2}\| L_{\theta}^{*}v\|_{(-1,-2)} \textit{ for all } v\in \widetilde{C}_{x}^{\infty}(\overline{\Omega}). \end{gather*} \end{proposition} \begin{proof} We first consider the case when $v\in\widetilde{C}^{\infty}(\overline{\Omega})$. Let $u\in C^{\infty}(\Omega)\cap H^{4}(\Omega)$ be the unique solution of \[ Pu=v \quad\text{in } \Omega, \quad u(-x_{0},y)=u_{x}(-x_{0},y)=0, \quad u(x_{0},y)=0, \] given by Lemma \ref{lem2.4}. We now show that \begin{align*} &(Au+Bu_{x}+Cu_{y}+Du_{yy},L_{\theta}u)\\ &\geq C_{1}[\| u\|^{2}+\| Eu_{y}\|^{2} +\theta(\| u_{x}\|^{2}+\| u_{xy}\|^{2} +\| u_{yy}\|^{2}+\theta\| u_{xyy}\|^{2})], \end{align*} where $A,B,C,D,E$, and $C_{1}$ were given in Lemma \ref{lem2.3}. Let $\nu_{k}$ be given by \eqref{e2.7} and define the sequence $\{u^{k}\}_{k=1}^{\infty}$, where $u^{k}=\nu_{k}u$. Then as in the proof of Lemma \ref{lem2.4} we have: \begin{itemize} \item[(i)] $u^{k}\in C^{\infty}(\Omega)\cap H^{4}(\Omega)$, \item[(ii)] $u^{k}$ has bounded support and $u^{k}_{x}(-x_{0},y)=0$, $u^{k}(x_{0},y)=0$, \item[(iii)] $\| u-u^{k}\|_{4}\to 0$ as $k\to\infty$, \item[(iv)] $\| Eu_{y}-Eu^{k}_{y}\|\to 0$ as $k\to\infty$. \end{itemize} Let $\{u_{k}\}_{k=1}^{\infty}$ a $C^{\infty}$ approximation of $\{u^{k}\}_{k=1}^{\infty}$ such that: \begin{itemize} \item[(i)] $u_{k}\in C^{\infty}(\overline{\Omega})$, \item[(ii)] $u_{k}$ has bounded support and $(u_{k})_{x}(-x_{0},y)=0$, $u_{k}(x_{0},y)=0$, \item[(iii)] $\| u^{k}-u_{k}\|_{4}\to 0$ as $k\to\infty$, \item[(iv)] $\| Eu^{k}_{y}-E(u_{k})_{y}\|\to 0$ as $k\to\infty$. \end{itemize} Then applying Lemma \ref{lem2.3} we have \begin{align*} &(Au+Bu_{x}+Cu_{y}+Du_{yy},L_{\theta}u) \\ &=\lim_{k\to\infty}(Au_{k}+B(u_{k})_{x} +C(u_{k})_{y}+D(u_{k})_{yy},L_{\theta}u_{k}) \\ & \geq\lim_{k\to\infty} C_{1}[\|u_{k} \|^{2} + \|E(u_{k})_{y} \|^{2} +\theta(\|(u_{k})_{x} \|^{2} \\ &\quad + \|(u_{k})_{xy} \|^{2} + \|(u_{k})_{yy} \|^{2} + \theta\|(u_{k})_{xyy} \|^{2})] \\ &=C_{1}[\| u\|^{2} +\| Eu_{y}\|^{2} +\theta(\| u_{x}\|^{2}+\| u_{xy}\|^{2} +\| u_{yy}\|^{2}+\theta\| u_{xyy}\|^{2})]. \end{align*} By the above estimate and definition of the negative norms, it follows that \begin{align*} \| L_{\theta}^{*}v\|_{(-1,-2)}\| u\| _{(1,2)} &\geq (L_{\theta}^{*}v,u)\\ &=(v,L_{\theta}u)\\ &=(Au+Bu_{x}+Cu_{y}+Du_{yy},L_{\theta}u)\\ &\geq C_{1}[\| u\|^{2} +\|Eu_{y}\|^{2} +\theta(\| u_{x}\|^{2}+\|u_{xy}\|^{2} +\| u_{yy}\|^{2}\\ &\quad +\theta\|u_{xyy}\|^{2})]. \end{align*} Furthermore using Cauchy's inequality and the equation $Pu=v$, we obtain \begin{align*} \| L_{\theta}^{*}v\|_{(-1,-2)} &\geq C_{1}'[\| u\|+\| Eu_{y}\| +\sqrt{\theta}(\| u_{x}\|+\|u_{xy}\|+\| u_{yy}\|+\sqrt{\theta}\| u_{xyy}\|)]\\ &\geq M_{1}^{-1}\| v\|, \end{align*} for some constants $C_{1}',M_{1}>0$. Moreover, similar arguments may be used to treat the case when $v\in \widetilde{C}_{x}^{\infty}(\overline{\Omega})$. \end{proof} The existence of weak solutions to problems \eqref{e2.3} and \eqref{e2.4} immediately follows from Proposition \ref{prop2.1} by a standard functional analytic argument. We include the proof here for convenience. \begin{theorem} \label{thm2.1} For each $f\in L^{2}(\Omega)$ there exists a weak solution $u\in \widetilde{H}^{(1,2)}(\Omega)$, $\widetilde{H}_{x}^{(1,2)}(\Omega)$ of \eqref{e2.3}, \eqref{e2.4} respectively. \end{theorem} \begin{proof} We shall first treat problem \eqref{e2.3}. Let $W= L_{\theta}^{*}(\widetilde{C}^{\infty}(\overline{\Omega}))$ and define the linear functional $F:W\to\mathbb{R}$ by \[ F(L_{\theta}^{*}v)=(f,v). \] Using Proposition \ref{prop2.1}, the following calculation will show that $F$ is bounded as a linear functional on the subspace $W$ of $\widetilde{H}^{(-1,-2)}(\Omega)$, \[ |F(L_{\theta}^{*}v)|= |(f,v)| \leq \| f\|\| v\| \leq M_{1}\| f\|\| L_{\theta}^{*}v \|_{(-1,-2)}. \] Use the Hahn-Banach theorem to extend $F$ from $W$ to the whole space $\widetilde{H}^{(-1,-2)}(\Omega)$. It follows from the Riesz representation theorem that there exists $u\in \widetilde{H}^{(1,2)}(\Omega)$ such that \[ F(w)=(u,w) \text{ for all } w\in\widetilde{H}^{(-1,-2)}(\Omega). \] Thus, restricting $w$ to $W$ we have \[ (u,L_{\theta}^{*}v)=F(L_{\theta}^{*}v)=(f,v) \text{ for all } v\in\widetilde{C}^{\infty}(\overline{\Omega}). \] The case of problem \eqref{e2.4} may be treated in a similar manner. \end{proof} We now prove the existence of weak solutions for the adjoint problems \eqref{e2.5} and \eqref{e2.6}. The existence of solutions for these problems will be needed in the next section, where they will aid in proving higher regularity for solutions of \eqref{e2.3}. The formal adjoint of $L_{\theta}$ is given by \begin{align*} L_{\theta}^{*} &=-\theta\partial_{xxyy}+a_{11}\partial_{xx} +a_{22}\partial_{yy} +(2\partial_{x}a_{11}-a_{1})\partial_{x} \\ &\quad +(2\partial_{y}a_{22}-a_{2})\partial_{y}+(\partial_{xx}a_{11}+ \partial_{yy}a_{22}-\partial_{x}a_{1}-\partial_{y}a_{2}+a). \end{align*} All the coefficients of $L_{\theta}^{*}$, denoted $a_{ij}^{*},a_{i}^{*},a^{*}$, have the same properties as the coefficients of $L_{\theta}$, except $a_{2}^{*}=2\partial_{y}a_{22}-a_{2}$. This difference will not allow us to directly apply the above procedure to obtain weak solutions for \eqref{e2.5} and \eqref{e2.6}. However if \[ h(x,y)=e^{2\int_{0}^{y}\frac{a_{2}(x,t)}{a_{22}(x,t)}dt}, \] then by setting $v=hw$, the equation $L_{\theta}^{*}v=g$ becomes $\overline{L}_{\theta}^{*}w=g/h$, where \begin{align*} \overline{L}_{\theta}^{*} &=-\theta\partial_{xxyy}-2\theta\frac{h_{y}}{h} \partial_{xxy}-2\theta\frac{h_{x}}{h}\partial_{xyy} \\ &\quad +(a^{*}_{11}-\theta \frac{h_{yy}}{h})\partial_{xx}-4\theta\frac{h_{xy}}{h}\partial_{xy} +(a^{*}_{22}-\theta\frac{h_{xx}}{h})\partial_{yy} \\ &\quad +(a^{*}_{2}+2a^{*}_{22}\frac{h_{y}}{h}-2\theta\frac{h_{xxy}}{h})\partial_{y} +(a^{*}_{1}+2a^{*}_{11}\frac{h_{x}}{h}-2\theta\frac{h_{xyy}}{h})\partial_{x} \\ &\quad +(a^{*}_{11}\frac{h_{xx}}{h}+a^{*}_{22}\frac{h_{yy}}{h}+a^{*}_{1}\frac{h_{x}}{h} +a^{*}_{2}\frac{h_{y}}{h}+a^{*}-\theta\frac{h_{xxyy}}{h}). \end{align*} The special choice of $h$ guarantees that the coefficient of $\partial_{y}$ in $\overline{L}_{\theta}^{*}$ is $3a_{2}+O( \varepsilon+\theta)$, so that all the coefficients of $\overline{L}_{\theta}^{*}$ have the same properties as the coefficients of $\overline{L}_{\theta}$, where $\overline{L}_{\theta}w=f/h$ is the equation obtained from $L_{\theta}u=f$ by setting $u=hw$. Therefore if $g\in L^{2}(\Omega)$, the problems \begin{gather*} \overline{L}_{\theta}^{*}w=g/h \quad\text{in } \Omega, \quad w|_{\partial\Omega}=0, \\ \overline{L}_{\theta}^{*}w=g/h \quad \text{in } \Omega, \quad w_{x}|_{\partial\Omega}=0, \end{gather*} have weak solutions of the form $w=v/h$, where $v\in\widetilde{H}^{(1,2)}(\Omega)$, $\widetilde{H}_{x}^{(1,2)}(\Omega)$ respectively. We then obtain the following result. \begin{corollary} \label{coro2.1} For each $g\in L^{2}(\Omega)$ there exists a weak solution $v\in\widetilde{H}^{(1,2)}(\Omega)$, $\widetilde{H}_{x}^{(1,2)}(\Omega)$ of \eqref{e2.5}, \eqref{e2.6} respectively. \end{corollary} \section{Linear Regularity} The purpose of this section is to establish the regularity in $X$, of weak solutions to problem \eqref{e2.3} for a particular choice of the right-hand side, $f$. This shall be accomplished by establishing the uniqueness of weak solutions to problems \eqref{e2.3} and \eqref{e2.4} in $L^{2}(\Omega)$, and then applying a boot-strap argument. To obtain the uniqueness of weak solutions, we will utilize the notion of a strong solution, in particular, for first order systems. The definition of a strong solution will be given below. We first introduce the notation and terminology that will be used for first order systems. Consider a boundary value problem \begin{equation} \label{e3.1} SU=A_{1}U_{x}+A_{2}U_{y}+A_{3}U=F \quad \text{in } \Omega, \quad U|_{\partial\Omega}\in N, \end{equation} where $A_{1},A_{2},A_{3}$ are $n\times n$ matrices, $U$ and $F$ are $n$-vectors, and $N$ is a linear subspace of the space of $n$-vector valued functions restricted to $\partial\Omega$. The corresponding adjoint problem is given by \[ S^{*}V=-A^{*}_{1}V_{x}-A^{*}_{2}V_{y}+(A^{*}_{3}-\partial_{x}A^{*}_{1} -\partial_{y}A^{*}_{2})V=G \quad \text{in } \Omega, \quad V|_{\partial\Omega}\in N^{*}, \] where $A^{*}_{i}$ denotes the transpose of $A_{i}$, and $N^{*}$ is the orthogonal complement of $\bigtriangleup N$, where $\bigtriangleup$ is the matrix defined on $\partial\Omega$ by $A_{1}n_{1}+A_{2}n_{2}$, and $(n_{1},n_{2})$ is the unit outward normal to $\partial\Omega$. Let $F\in L^{2}(\Omega)$. The notion of a weak solution to problem \eqref{e3.1} is similar to the definition given in section $\S 2$ for single equations. That is, $U\in L^{2}(\Omega)$ is said to be a weak solution of \eqref{e3.1} whenever \[ (S^{*}V,U)=(V,F), \] for every $V\in C^{\infty}(\overline{\Omega})$ with bounded support and such that $V|_{\partial\Omega}\in N^{*}$. We now give the definition of a strong solution. \begin{definition} \label{def3.1} $U\in L^{2}(\Omega)$ is a strong solution of \eqref{e3.1} if there exists a sequence $\{U_{k}\}^{\infty}_{k=1}$, such that $U_{k}\in C^{\infty}(\overline{\Omega})$ with bounded support, $U_{k}|_{\partial\Omega}\in N$, and \[ \| U_{k}-U\|\to 0, \quad \| SU_{k}-F\|\to 0, \quad \textit{as } k\to\infty. \] \end{definition} Clearly a strong solution is a weak solution. Moreover, using techniques developed by Friedrichs \cite{f1} and Lax/Phillips \cite{l2}, Peyser \cite{p4} has obtained the following converse statement. \begin{theorem}[Identity of Weak and Strong Solutions] \label{thm3.1} Let the following conditions on the operator $S$ and the boundary space $N$ be satisfied: \begin{itemize} \item[(i)] The matrix $\bigtriangleup$ is of constant rank in a neighborhood of the boundary, \item[(ii)] $N$ is of constant dimension at each point of the boundary, \item[(iii)] $N$ contains the nullspace of $\bigtriangleup$. \end{itemize} Then a weak solution $U\in L^{2}(\Omega)$ of \eqref{e3.1} is also a strong solution. \end{theorem} Note that for our particular domain $\bigtriangleup=A_{1}n_{1}$, so that condition (i) is equivalent to $A_{1}$ having constant rank in a neighborhood of $\partial\Omega$. With the aim of applying Theorem \ref{thm3.1}, we shall transform problems \eqref{e2.3}, \eqref{e2.4}, \eqref{e2.5}, and \eqref{e2.6} into the setting of first order systems. Let $f,g\in L^{2}(\Omega)$ be the right-hand sides of \eqref{e2.3}, \eqref{e2.4} and \eqref{e2.5}, \eqref{e2.6} respectively, and define $A_{1}$, $\widetilde{A}_{1}$, $A_{2}$, $\widetilde{A}_{2}$, $A_{3}$, $\widetilde{A}_{3}$, $F$, and $G$ by \begin{gather*} A_{1}=\widetilde{A}_{1}=\begin{pmatrix} -\theta & 0 & a_{11} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \end{pmatrix}, \quad A_{2}=\widetilde{A}_{2}=\begin{pmatrix} 0 & 0 & 0 & a_{22} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \\ A_{3}=\begin{pmatrix} 0 & 0 & a_{1} & a_{2} & a \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \end{pmatrix}, \quad \widetilde{A}_{3}=\begin{pmatrix} 0 & 0 & a^{*}_{1} & a^{*}_{2} & a^{*} \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \end{pmatrix}, \\ F=\begin{pmatrix} f \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \quad G=\begin{pmatrix} g \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}. \end{gather*} Define boundary spaces \begin{gather*} N_{1}=\{(u_{1},\dots,u_{5})|_{\partial\Omega}: u_{5} |_{\partial\Omega}=0\}, \\ N_{2}=\{(u_{1},\dots,u_{5})|_{\partial\Omega}: (-\theta u_{1}+a_{11}u_{3})|_{\partial\Omega}=0\}. \end{gather*} Furthermore define boundary value problems \begin{gather} S_{\theta}U=A_{1}U_{x}+A_{2}U_{y}+A_{3}U=F \quad \text{in } \Omega, \quad U|_{\partial\Omega}\in N_{1}, \label{e3.2} \\ S_{\theta}U=F \quad \text{in } \Omega, \quad U|_{\partial\Omega}\in N_{2}, \label{e3.3} \\ \widetilde{S}_{\theta}V=\widetilde{A}_{1}V_{x}+\widetilde{A}_{2}V_{y} +\widetilde{A}_{3}V=G \quad \text{in } \Omega, \quad V|_{\partial\Omega}\in N_{1}, \label{e3.4} \\ \widetilde{S}_{\theta}V=G \quad \text{in } \Omega, \quad V|_{\partial\Omega}\in N_{2}. \label{e3.5} \end{gather} We now show that the weak solutions of \eqref{e2.3}, \eqref{e2.4}, \eqref{e2.5}, and \eqref{e2.6} given by Theorem \ref{thm2.1} and Corollary \ref{coro2.1} are also weak solutions of \eqref{e3.2}, \eqref{e3.3}, \eqref{e3.4}, and \eqref{e3.5} respectively. \begin{lemma} \label{lem3.1} Let $u\in\widetilde{H}^{(1,2)}(\Omega)$, $\widetilde{H}^{(1,2)}_{x}(\Omega)$ be a weak solution of \eqref{e2.3}, \eqref{e2.4} respectively, then $U=(u_{xyy},u_{yy},u_{x},u_{y},u)\in L^{2}(\Omega)$ is a weak solution of \eqref{e3.2}, \eqref{e3.3} respectively. Similarly if $v\in\widetilde{H}^{(1,2)}(\Omega)$, $\widetilde{H}^{(1,2)}_{x}(\Omega)$ is a weak solution of \eqref{e2.5}, \eqref{e2.6} respectively, then $V=(v_{xyy},v_{yy},v_{x},v_{y},v)\in L^{2}(\Omega)$ is a weak solution of \eqref{e3.4}, \eqref{e3.5} respectively. \end{lemma} \begin{proof} Let $u\in\widetilde{H}^{(1,2)}(\Omega)$ be a weak solution of problem \eqref{e2.3}. We will show that \begin{equation} \label{e3.6} \iint_{\Omega}U^{*}S^{*}_{\theta}V=\iint_{\Omega}F^{*}V \end{equation} for all $V\in C^{\infty}(\overline{\Omega})$ with bounded support such that $V|_{\partial\Omega}\in N^{*}_{1}$, where \[ N^{*}_{1}=\{(v_{1},\dots,v_{5})|_{\partial\Omega}: v_{1}|_{\partial\Omega}=v_{5}|_{\partial\Omega}=0\}. \] A calculation shows that \begin{equation} \label{e3.7} \begin{aligned} \iint_{\Omega}U^{*}S^{*}_{\theta}V & = \iint_{\Omega}(\theta u_{xyy}-a_{11}u_{x})\partial_{x}v_{1}-a_{22}u_{y}\partial_{y}v_{1} -(u\partial_{x}v_{2}+u_{x}v_{2})\\ & \quad +[(a_{1}-\partial_{x}a_{11})u_{x}+(a_{2}-\partial_{y}a_{22})u_{y}+au]v_{1}\\ &\quad -(u\partial_{y}v_{3}+u_{y}v_{3})-(u_{y}\partial_{y}v_{4} +u_{yy}v_{4})-(u_{yy}\partial_{x}v_{5}+u_{xyy}v_{5}). \end{aligned} \end{equation} Since $V|_{\partial\Omega}\in N^{*}_{1}$ and $u\in\widetilde{H}^{(1,2)}(\Omega)$ is a weak solution of \eqref{e2.3}, we can integrate by parts to obtain \[ \iint_{\Omega}U^{*}S^{*}_{\theta}V=\iint_{\Omega}uL^{*}_{\theta}v_{1}= \iint_{\Omega}fv_{1}=\iint_{\Omega}F^{*}V, \] showing that $U$ is a weak solution of \eqref{e3.2}. Let $u\in\widetilde{H}_{x}^{(1,2)}(\Omega)$ be a weak solution of \eqref{e2.4}. We now show that \eqref{e3.6} holds for all $V\in C^{\infty}(\overline{\Omega})$ with bounded support such that $V|_{\partial\Omega}\in N^{*}_{2}$, where \[ N^{*}_{2}=\{(v_{1},\dots,v_{5})|_{\partial\Omega}: v_{2}|_{\partial\Omega}=v_{5}|_{\partial\Omega}=0\}. \] From \eqref{e3.7} it follows that \begin{equation} \label{e3.8} \begin{aligned} \iint_{\Omega}U^{*}S^{*}_{\theta}V & = \iint_{\Omega}(\theta u_{xyy}-a_{11}u_{x})\partial_{x}v_{1}-a_{22}u_{y}\partial_{y}v_{1}\\ & \quad +[(a_{1}-\partial_{x}a_{11})u_{x}+(a_{2}- \partial_{y}a_{22})u_{y}+au]v_{1}. \end{aligned} \end{equation} To integrate by parts we construct an approximating sequence $\{v_{1}^{k}\}_{k=1}^{\infty}$ for $v_{1}$, such that $v_{1}^{k}\in\widetilde{C}_{x}^{\infty}(\overline{\Omega})$ and \[ \| v_{1}^{k}-v_{1}\|+\|\partial_{x} v_{1}^{k}-v_{1}\|\to 0 \quad \text{as } k\to\infty. \] Take a sequence $\{v_{k}\}_{k=1}^{\infty}\subset\widetilde{C}^{\infty}(\overline{\Omega})$ with the property that $\|v_{k}-\partial_{x}v_{1}\|\to 0$ as $k\to\infty$, and define \[ v^{k}_{1}=\int_{-x_{0}}^{x}v_{k}(t,y)dt+v_{1}(-x_{0},y). \] Then since \begin{align*} (v_{1}^{k}-v_{1})^{2}&= \Big(\int_{-x_{0}}^{x}\partial_{t} (v_{1}^{k}(t,y)-v_{1}(t,y))dt\Big)^{2}\\ &\leq 2x_{0}\int_{-x_{0}}^{x}(\partial_{t} v_{1}^{k}(t,y)-\partial_{t}v_{1}(t,y))^{2}dt, \end{align*} we have \[ \iint_{\Omega}(v_{1}^{k}-v_{1})^{2} \leq 4x_{0}^{2}\iint_{\Omega}(\partial_{x}v_{1}^{k} -\partial_{x}v_{1})^{2} =4x_{0}^{2}\iint_{\Omega}(v_{k}-\partial_{x}v_{1})^{2}\to 0, \] so that $v_{1}^{k}$ satisfies the desired properties. Recalling that $a_{1}|_{\partial\Omega}=$ $\partial_{x}a_{11}|_{\partial\Omega}=0$ by (ii) of Lemma \ref{lem2.2}, and using the fact that $u$ is a weak solution of \eqref{e2.4}, we can integrate by parts in \eqref{e3.8} to obtain \[ \iint_{\Omega}U^{*}S^{*}_{\theta}V = \lim_{k\to\infty}\iint_{\Omega} uL^{*}_{\theta}v_{1}^{k} = \lim_{k\to\infty}\iint_{\Omega} fv_{1}^{k} =\iint_{\Omega}fv_{1}=\iint_{\Omega}F^{*}V, \] showing that $U$ is a weak solution of \eqref{e3.3}. Similar arguments show that if $v\in\widetilde{H}^{(1,2)}(\Omega)$, $\widetilde{H}^{(1,2)}_{x}(\Omega)$ is a weak solution of \eqref{e2.5}, \eqref{e2.6} respectively, then $V=(v_{xyy},v_{yy},v_{x},v_{y},v)\in L^{2}(\Omega)$ is a weak solution of \eqref{e3.4}, \eqref{e3.5} respectively. \end{proof} Now that the weak solutions of the previous section have been translated into the setting of first order systems, Theorem \ref{thm3.1} is applicable. As a result, we obtain \begin{proposition} \label{prop3.1} The weak solutions of problems \eqref{e2.3} and \eqref{e2.4}, given by Theorem \ref{thm2.1}, are unique in $L^{2}(\Omega)$. \end{proposition} \begin{proof} Let $u\in\widetilde{H}^{(1,2)}(\Omega)$ be a weak solution of problem \eqref{e2.3} with $f=0$, then \begin{equation} \label{e3.9} (L^{*}_{\theta}w,u)=0 \quad \text{for all } w\in\widetilde{C}^{\infty}(\overline{\Omega}). \end{equation} We will show that $u=0$ in $L^{2}(\Omega)$. Let $v\in\widetilde{H}^{(1,2)}(\Omega)$ be the weak solution of \eqref{e2.5} with $g=u$. Then by Lemma \ref{lem3.1} $V=(v_{xyy},v_{yy},v_{x},v_{y},v)$ is a weak solution of \eqref{e3.4}. We now show that the conditions of Theorem \ref{thm3.1} are satisfied for problem \eqref{e3.4}. Condition (ii) is immediately satisfied, and since $a_{11}^{*}\leq-\theta$ in a neighborhood of $\partial\Omega$, condition (i) is satisfied with $\triangle=\pm\widetilde{A}_{1}$ having the constant rank of 3. Furthermore the nullspace of $\triangle$ is given by \[ \{(v_{1},\dots,v_{5})|_{\partial\Omega}\mid(-\theta v_{1}+a_{11}v_{3})|_{\partial\Omega}=v_{2}|_{\partial\Omega}=v_{5} |_{\partial\Omega}=0\}, \] which is contained in $N_{1}$ so that condition (iii) is satisfied. Therefore we can apply Theorem \ref{thm3.1} to obtain an approximating sequence $\{V_{k}\}_{k=1}^{\infty}$ for $V$, such that $V_{k}\in C^{\infty}(\overline{\Omega})$ with bounded support, $V_{k}|_{\partial\Omega}\in N_{1}$, and \begin{equation} \label{e3.10} \| V_{k}-V\|\to 0, \quad \| \widetilde{S}_{\theta}V_{k}-G\|\to 0 \quad \text{as } k\to\infty. \end{equation} From \eqref{e3.10} it follows that \begin{gather*} \|v_{k}^{1}-v_{xyy}\|\to 0, \quad \| v_{k}^{2}-v_{yy}\|\to 0, \quad \|v_{k}^{3}-v_{x}\| \to 0, \\ \| v_{k}^{4}-v_{y}\|\to 0, \quad \| v_{k}^{5}-v\|\to 0, \\ \|(-\theta\partial_{x}v_{k}^{1}+a_{11}^{*}\partial_{x} v_{k}^{3}+a_{22}^{*}\partial_{y}v_{k}^{4}+a_{1}^{*}v_{k}^{3} +a_{2}^{*}v_{k}^{4}+a^{*}v_{k}^{5})-u\|\to 0. \end{gather*} Hence \begin{align*} (u,u) &=\lim_{k\to\infty}\iint_{\Omega}[ -\theta\partial_{x}v_{k}^{1}+a_{11}^{*}\partial_{x}v_{k}^{3} +a_{22}^{*}\partial_{y}v_{k}^{4}+a_{1}^{*}v_{k}^{3}+a_{2}^{*}v_{k}^{4} +a^{*}v_{k}^{5}]u \\ &=\lim_{k\to\infty}\iint_{\Omega}(\theta v_{k}^{1} - a_{11}^{*}v_{k}^{3})u_{x} - a_{22}^{*}v_{k}^{4}u_{y} + [(a_{1}^{*} - \partial_{x}a_{11}^{*})v_{k}^{3} \\ &\quad + (a_{2}^{*} - \partial_{y}a_{22}^{*}) v_{k}^{4} + a^{*}v_{k}^{5}]u \\ &=\iint_{\Omega}(\theta v_{xyy}-a_{11}^{*}v_{x})u_{x}-a_{22}^{*}v_{y}u_{y}+ [(a_{1}^{*}-\partial_{x}a_{11}^{*})v_{x}\\ &\quad +(a_{2}^{*}-\partial_{y}a_{22}^{*}) v_{y}+a^{*}v]u. \end{align*} Let $\{v_{n}\}_{n=1}^{\infty}\subset \widetilde{C}^{\infty}(\overline{\Omega})$ be an approximating sequence for $v$ in $\widetilde{H}^{(1,2)}(\Omega)$. Then integrating by parts and using \eqref{e3.9}, we obtain \[ (u,u)=\lim_{n\to\infty}\iint_{\Omega}(L_{\theta}^{*}v_{n})u=0. \] Similar arguments hold for problem \eqref{e2.4} \end{proof} Having established the uniqueness of weak solutions, we are now ready to apply a boot-strap procedure to obtain higher regularity for problem \eqref{e2.3} in the $x$-direction. \begin{theorem} \label{thm3.2} Let $u$ and $f$ be as in problem \eqref{e2.3}. Let $s\leq r-4$ and $f\in H^{s}(\Omega)$ be such that $\partial_{x}^{\alpha}f|_{\partial\Omega}=0$ for $\alpha\leq s-1$. If $\varepsilon=\varepsilon(s)$ is sufficiently small, then for all $\alpha\leq s$, $\partial_{x}^{\alpha}u\in\widetilde{H}^{(1,2)}(\Omega)$ when $\alpha$ is even, and $\partial_{x}^{\alpha}u\in \widetilde{H}_{x}^{(1,2)}(\Omega)$ when $\alpha$ is odd. \end{theorem} \begin{proof} The case $s=0$ is given by Theorem \ref{thm2.1}. Consider the case $s=1$. Let $w=u_{x}$ and formally differentiate the equation $L_{\theta}u=f$ with respect to $x$: \begin{align*} L_{1}w&:=-\theta w_{xxyy}+a_{11}w_{xx}+a_{22}w_{yy}+(a_{1}+\partial_{x}a_{11})w_{x} +a_{2}w_{y}+(a+\partial_{x}a_{1})w\\ &= f_{x}-u_{yy}\partial_{x}a_{22}-u_{y}\partial_{x}a_{2}-u\partial_{x}a :=f_{1}. \end{align*} Observe that since $\partial_{x}a_{11},\partial_{x}a_{1}=O(\varepsilon)$ and both vanish outside $X$, the operator $L_{1}$ has the same existence and uniqueness properties as $L_{\theta}$. Furthermore, by restricting $L_{\theta}u=f$ to the boundary of $\Omega$ and using $u|_{\partial\Omega}=a_{1}|_{\partial\Omega}=0$, we obtain the following ODE \begin{equation} \label{e3.11} (-\theta u_{xxyy}+a_{11}u_{xx})|_{\partial\Omega}=0, \end{equation} for which the only solution in $L^{2}(\partial\Omega)$ is $u_{xx}|_{\partial\Omega}=0$. Therefore, in the regular case $w=u_{x}$ satisfies problem \eqref{e2.4} with $L_{\theta}$ and $f$ replaced by $L_{1}$ and $f_{1}$. Let $u\in\widetilde{H}^{(1,2)}(\Omega)$ be the weak solution of problem \eqref{e2.3}. We now show that $u_{x}\in L^{2}(\Omega)$ is a weak solution of \eqref{e2.4} with $L_{\theta}$ and $f$ replaced by $L_{1}$ and $f_{1}\in L^{2}(\Omega)$; we denote this problem by $\text{\eqref{e2.4}}_{1}$. Let $v\in\widetilde{C}^{\infty}_{x}(\overline{\Omega})$, then \begin{align*} &(u_{x},L_{1}^{*}v) \\ &= -(u,(L_{1}^{*}v)_{x})=-(u,L^{*}(v_{x})) +(u,L^{*}(v_{x})-(L_{1}^{*}v)_{x})\\ &= -(f,v_{x})+(u,-v_{yy}\partial_{x}a_{22} +v_{y}[ \partial_{x}a_{2}-2\partial_{xy}a_{22}] +v[-\partial_{x}a - \partial_{xyy}a_{22}+\partial_{xy}a_{2}])\\ &= (f_{x},v)+(-u_{yy}\partial_{x}a_{22}-u_{y}\partial_{x}a_{2} -u\partial_{x}a,v)=(f_{1},v). \end{align*} Therefore $u_{x}$ is a weak solution of $\text{\eqref{e2.4}}_{1}$, and by the uniqueness result Proposition \ref{prop3.1}, $u_{x}$ must coincide with the solution in $\widetilde{H}_{x}^{(1,2)}(\Omega)$ given by Theorem \ref{thm2.1}. Hence $u_{x}\in\widetilde{H}_{x}^{(1,2)}(\Omega)$. We now consider the case $s=2$. Let $w=u_{xx}$ and formally differentiate the equation $L_{1}u_{x}=f_{1}$ with respect to $x$: \begin{align*} L_{2}w&:=-\theta w_{xxyy}+a_{11}w_{xx}+a_{22}w_{yy}\\ &\quad +(a_{1}+2\partial_{x}a_{11})w_{x} +a_{2}w_{y}+(a+2\partial_{x}a_{1}+\partial_{xx}a_{11})w\\ &=\partial_{x}f_{1}-u_{xyy}\partial_{x}a_{22}-u_{xy}\partial_{x}a_{2}-u_{x} (\partial_{x}a+\partial_{xx}a_{1}) :=f_{2}. \end{align*} Again since $\partial_{x}a_{11},\partial_{xx}a_{11},\partial_{x}a_{1}=O(\varepsilon)$ and all three vanish outside $X$, the operator $L_{2}$ has the same existence and uniqueness properties as $L_{\theta}$, provided that $\varepsilon$ is sufficiently small. Also, when $u$ is regular $u_{xx}|_{\partial\Omega}=0$ from \eqref{e3.11}. Thus in the regular case $w=u_{xx}$ satisfies \eqref{e2.3} with $L_{\theta}$ and $f$ replaced by $L_{2}$ and $f_{2}\in L^{2}(\Omega)$; we denote this problem by \eqref{e2.3}$_2$. Let $u\in\widetilde{H}^{(1,2)}(\Omega)$ be the weak solution of \eqref{e2.3}, then we know that $u_{x}\in\widetilde{H}_{x}^{(1,2)}(\Omega)$. We now show that $u_{xx}\in L^{2}(\Omega)$ is a weak solution of $\text{\eqref{e2.3}}_{2}$. Note that $L_{\theta}u=f$ in $L^{2}(\Omega)$ and let $v\in\widetilde{C}^{\infty}(\overline{\Omega})$, then a calculation produces \begin{align*} (u_{xx},L_{2}^{*}v) &=(u_{xxyy},-\theta v_{xx})+(u_{xx},(a_{11}v)_{xx})+(u_{yy},(a_{22}v)_{xx})+(u_{y},(a_{2}v)_{xx})\\ & \quad +(u_{x},[(a_{1}+2\partial_{x}a_{11})v]_{xx}) +(u,[(a+2\partial_{x}a_{1}+\partial_{xx}a_{11})v]_{xx})\\ &=(L_{\theta}u,v_{xx})+(f_{2}-f_{xx},v)=(f,v_{xx})+(f_{2}-f_{xx},v) =(f_{2},v). \end{align*} By the uniqueness of weak solutions for problem $\text{\eqref{e2.3}}_{2}$, $u_{xx}$ must coincide with the solution in $\widetilde{H}^{(1,2)}(\Omega)$. Thus $u_{xx}\in\widetilde{H}^{(1,2)}(\Omega)$. To obtain the regularity of higher order derivatives, we observe that the above procedure applied to $L_{\theta}u=f$ holds for $L_{2}u_{xx}=f_{2}$, since for $\alpha\geq 1$ \[ \partial_{x}^{\alpha}a_{11}|_{\partial\Omega}= \partial_{x}^{\alpha}a_{22}|_{\partial\Omega}= \partial_{x}^{\alpha}a_{i}|_{\partial\Omega}= \partial_{x}^{\alpha}a|_{\partial\Omega}=0, \] so that $f_{2}|_{\partial\Omega}=0$. Therefore $u_{xxx}\in \widetilde{H}_{x}^{(1,2)}(\Omega)$ and $u_{xxxx}\in\widetilde{H}^{(1,2)}(\Omega)$. Furthermore, we can continue this process until $f$ and the coefficients of $L_{\theta}$ run out of derivatives, as long as $\varepsilon$ is chosen sufficiently small depending on the size of $s$. \end{proof} We now prove regularity in the $y$-direction for the weak solution of problem \eqref{e2.3}. The following standard lemma concerning difference quotients will be needed. \begin{lemma} \label{lem3.2} Let $w\in L^{2}(\Omega)$ have bounded support, and define \[ w^{h}=\frac{1}{h}(w(x,y+h)-w(x,y)). \] If $\| w^{h}\|\leq M$ where $M$ is independent of $h$, then $w\in H^{(0,1)}(\Gamma)$ for any compact $\Gamma\subset\Omega$. Furthermore, if $w\in H^{(0,1)}(\Omega)$ then $\| w^{h}\|\leq M\| w_{y}\|$. \end{lemma} \begin{theorem} \label{thm3.3} Let the hypotheses of Theorem \ref{thm3.2} hold, then $u\in H^{s}(X)$. \end{theorem} \begin{proof} From Theorem \ref{thm3.2} we know that $\partial_{x}^{\alpha}u\in H^{(1,2)}(\Omega)$ for $0\leq\alpha\leq s$. Therefore the following equality holds in $L^{2}(\Omega)$, \begin{equation} \label{e3.12} \widetilde{L}u_{yy}:=-\theta u_{xxyy}+a_{22}u_{yy}= f-a_{11}u_{xx}-a_{1}u_{x}-a_{2}u_{y}-au:=\widetilde{f}. \end{equation} Since $|a_{2}|=O(|y|)$ as $|y|\to\infty$, we do not necessarily know that $\widetilde{f}\in H^{(0,1)}(\Omega)$; however, we do have $\widetilde{f}\in H^{(0,1)}(\Gamma)$ for any compact $\Gamma\subset\Omega$. Fix a constant $k>y_{0}$ and set $w=\nu_{k}u_{yy}$, where $\nu_{k}$ is given by \eqref{e2.7}. Then \begin{equation} \label{e3.13} \widetilde{L}w^{h}=(\nu_{k}\widetilde{f})^{h} -\nu_{k}(y+h)u_{yy}(x,y+h)a_{22}^{h}. \end{equation} Since $u\in\widetilde{H}^{(1,2)}(\Omega)$, by multiplying \eqref{e3.13} on both sides by $w^{h}$ and integrating by parts we obtain \[ \| w^{h}\|+\| w_{x}^{h}\| \leq M_{1}(\| (\nu_{k}\widetilde{f})^{h}\|+1), \] for some $M_{1}$ independent of $h$. By Lemma \ref{lem3.2} \[ \| w^{h}\|+\| w_{x}^{h}\| \leq M_{2}(\|\nu_{k}\widetilde{f}\|_{(0,1)}+1), \] independent of $h$. Therefore $w_{y}$, $w_{xy}\in L^{2}(X)$, which implies that $\partial_{y}^{3}u$, $\partial_{x}\partial_{y}^{3}u\in L^{2}(X)$. Furthermore by differentiating $L_{\theta}u=f$ with respect to $x$, $\alpha=1,\dots,s-3$ times, the same procedure yields $\partial_{x}^{\alpha}\partial_{y}^{3}u\in L^{2}(X)$. Proceeding by induction on $l$, assume that $\partial_{x}^{\alpha}\partial_{y}^{\beta}u\in L^{2}(X)$, $\alpha\leq s-\beta$, $\beta\leq l$, and $3\leq l0$, and $C_{s}$ is a constant independent of $\varepsilon$ and $\theta$. Estimate \eqref{e4.1} will first be established in the coordinates $(\xi,\eta)$, which we have been denoting by $(x,y)$ for convenience, and later converted into the original coordinates $(x,y)$ of the introduction. We will need the following preliminary lemmas. The first is a modification of Lemma \ref{lem2.3}, and the second contains standard consequences of the interpolation inequalities for Sobolev spaces. \begin{lemma} \label{lem4.1} Let $w\in\widetilde{H}^{(2,2)}({\Omega})$ (or $\widetilde{H}_{x}^{(2,2)}({\Omega})$) be such that $yw\in L^{2}(\Omega)$, and let $p_{1}=\varepsilon\theta\widetilde{p_{1}}$, $p_{2}=\varepsilon\widetilde{p_{2}}$, $p_{3}=\varepsilon\widetilde{p_{3}}$, where $\widetilde{p_{i}}\in C^{\infty}_{c}(X)$, $i=1,2,3$. Then for $\varepsilon$ and $\theta$ sufficiently small, there exists a constant $M$ independent of $\varepsilon$ and $\theta$, such that \[ \| w\|+\| w_{y}\|\leq M\| p_{1}w_{xyy}+p_{2}w_{x}+p_{3}w+L_{\theta}w\|. \] \end{lemma} \begin{proof} Assume temporarily that $w\in \widetilde{C}^{\infty}(\overline{\Omega})$ (or $\widetilde{C}_{x}^{\infty}(\overline{\Omega})$). The properties of $p_{2}$ and $p_{3}$ guarantee that Lemma \ref{lem2.3} holds for the operator $p_{2}\partial_{x}+p_{3}+L_{\theta}$. Therefore \begin{equation} \label{e4.2} (Aw+Cw_{y}+Dw_{yy},p_{2}w_{x}+p_{3}w+L_{\theta}w)\geq \end{equation} \[ C_{1}[\| w\|^{2}+\| w_{y}\|^{2} +\theta(\| w_{x}\|^{2}+\| w_{xy}\|^{2}+\| w_{yy}\|^{2})] \] where $A,C,D$, and $C_{1}$ were given in Lemma \ref{lem2.3}. Furthermore integrating by parts yields \begin{equation} \label{e4.3} \begin{aligned} &(Aw+Cw_{y}+Dw_{yy},p_{1}w_{xyy}) \\ &=\iint_{\Omega}[-\frac{1}{2} (Dp_{1})_{x}]w_{yy}^{2}+[-Cp_{1}]w_{xy}w_{yy} +[\frac{1}{2}(Cp_{1})_{xy} +\frac{1}{2}(Ap_{1})_{x}]w_{y}^{2} \\ &\quad +[(Ap_{1})_{y}]w_{x}w_{y}+[-\frac{1}{2}(Ap_{1})_{xyy}]w^{2}. \end{aligned} \end{equation} All the boundary integrals vanish since $p_{1}\in C_{c}^{\infty}(X)$. Moreover the properties of $p_{1}$ guarantee that by choosing $\varepsilon$ and $\theta$ sufficiently small, we obtain the following by adding \eqref{e4.2} and \eqref{e4.3}, \begin{align*} &(Aw+Cw_{y}+Dw_{yy},p_{1}w_{xyy}+p_{2}w_{x}+p_{3}w+L_{\theta}w)\\ &\geq C_{1}[\| w\|^{2}+\| w_{y}\|^{2} +\theta(\| w_{x}\|^{2}+\| w_{xy}\|^{2}+\| w_{yy}\|^{2})]. \end{align*} Then an application of Cauchy's inequality, and the use of an approximating sequence $\{w_{k}\}_{k=1}^{\infty}$, as was constructed in Proposition \ref{prop2.1}, removes the assumption that $w\in \widetilde{C}^{\infty}(\overline{\Omega})$ (or $\widetilde{C}_{x}^{\infty}(\overline{\Omega})$) and completes the proof. \end{proof} \begin{lemma}[\cite{t1}] \label{lem4.2} Let $u,v\in H^{s}(X)$. \begin{itemize} \item[(i)] If $0\leq i\leq j \leq s$, then there exists a constant $\mathcal{M}_{i,j,s}$ such that \[ \| u\|_{H^{j}(X)}\leq \mathcal{M}_{i,j,s}\| u\| ^{\frac{s-j}{s-i}}_{H^{i}(X)}\| u\|^{\frac{j-i}{s-i}}_{H^{s}(X)}. \] \item[(ii)] If $\alpha$ and $\beta$ are multi-indices such that $|\alpha|+|\beta|=s$, then there exists a constant $\mathcal{M}_{s}$ such that \[ \|\partial^{\alpha}u\partial^{\beta}v\|_{L^{2}(X)}\leq \mathcal{M}_{s}(|u|_{L^{\infty}(X)}\| v\|_{H^{s}(X)}+ \| u\|_{H^{s}(X)}|v|_{L^{\infty}(X)}). \] \item[(iii)] Let $\Gamma\subset\mathbb{R}^{N}$ be compact and contain the origin, and let $G\in C^{\infty}(\Gamma)$. If $u\in H^{s+2}(X,\Gamma)$ and $\| u\| _{H^{2}(X)}\leq \mathcal{C}$ for some fixed $\mathcal{C}$, then there exists a constant $\mathcal{M}_{s}$ such that \[ \| G\circ u\|_{H^{s}(X)}\leq\mathrm{Vol}(X)|G(0)| +\mathcal{M}_{s}\| u\|_{H^{s+2}(X)}. \] \end{itemize} \end{lemma} Estimate \eqref{e4.1} will be established by induction on $s$, and we begin by estimating the $x$-derivatives. Let $\|\cdot\|_{s,X}$ denote $\|\cdot\| _{H^{s}(X)}$, and $|\cdot|_{\infty}$ denote $|\cdot|_{L^{\infty}(X)}$. \begin{proposition} \label{prop4.1} Let $u$ and $f$ be as in Theorem \ref{thm3.2}. If $\varepsilon=\varepsilon(s)$ is sufficiently small then \[ \|\partial^{s}_{x}u\|+\|\partial^{s}_{x}u_{y}\| \leq C_{s}(\| f\|_{s}+\| u\|_{s-1,X}+\Lambda_{s+2}\| f\|_{2,X}) \] for $s\leq r-6$, where $C_{s}$ is independent of $\varepsilon$ and $\theta$, and \[ \Lambda_{s+2}=\sum\| a_{ij}\|_{s+2,X}+\| a_{i}\|_{s+2,X}+\| a\|_{s+2,X}. \] \end{proposition} \begin{proof} We proceed by induction on $s$. The case $s=0$ follows from Lemma \ref{lem2.3}. Differentiate $L_{\theta}u=f$ $s$-times with respect to $x$ and put $w=\partial_{x}^{s}u$, then \begin{equation} \label{e4.4} \begin{aligned} &-\theta w_{xxyy}+a_{11}w_{xx}+a_{22}w_{yy}+(a_{1}+s\partial_{x}a_{11})w_{x} +a_{2}w_{y}+a_{s}w \\ &=\partial_{x}^{s}f-\sum_{i=0}^{s-1}\partial_{x}^{i}(\partial_{x} a_{22}\partial_{x}^{s-1-i}u_{yy})-\sum_{i=0}^{s-1}\partial_{x}^{i}(\partial_{x} a_{2}\partial_{x}^{s-1-i}u_{y}) \\ &\quad -\sum_{i=0}^{s-1}\partial_{x}^{i}(\partial_{x} a_{s-1-i}\partial_{x}^{s-1-i}u) :=f_{s} \end{aligned} \end{equation} where $a_{s}=a+s\partial_{x} a_{1}+\frac{s(s-1)}{2}\partial_{x}^{2}a_{11}$. A calculation shows that \begin{align*} &\sum_{i=0}^{s-1}\partial_{x}^{i}(\partial_{x} a_{22}\partial_{x}^{s-1-i}u_{yy})\\ &=s\partial_{x}a_{22} \partial_{x}^{s-1}u_{yy} +\frac{s(s-1)}{2}\partial_{x}^{2}a_{22} \partial_{x}^{s-2}u_{yy} +\sum_{i=2}^{s-1}\sum_{j=2}^{i} \begin{pmatrix}i\\j\end{pmatrix} \partial_{x}^{j+1}a_{22}\partial_{x}^{s-1-j}u_{yy}. \end{align*} Note that the term $\partial_{x}^{s-1}u_{yy}$ contains too many derivatives. However since $a_{22}=1+O(\varepsilon)$, we can solve for $\partial_{x}^{s-1}u_{yy}$ in \eqref{e4.4} with $s$ replaced by $s-1$ to obtain a more manageable expression: \[ \partial_{x}^{s-1}u_{yy}=\frac{1}{a_{22}}[\theta w_{xyy}-a_{11}w_{x}-(a_{1}+s\partial_{x}a_{11})w-a_{2}\partial_{x}^{s-1}u_{y} -a_{s-1}\partial_{x}^{s-1}u+f_{s-1}]. \] Substituting back into \eqref{e4.4}, we have \begin{align*} &\frac{s\theta\partial_{x}a_{22}}{a_{22}}w_{xyy}+(s\partial_{x}a_{11}- \frac{sa_{11}\partial_{x}a_{22}}{a_{22}})w_{x}\\ &+(a_{s}-a -\frac{s\partial_{x}a_{22}}{a_{22}}(a_{1}-s\partial_{x}a_{11}))w+L_{\theta}w \\ &=\partial_{x}^{s}f-\frac{s(s - 1)}{2}\partial_{x}^{2}a_{22} \partial_{x}^{s-2}u_{yy} -\sum_{i=2}^{s-1} \sum_{j=2}^{i} \begin{pmatrix}i\\j\end{pmatrix} \partial_{x}^{j+1}a_{22}\partial_{x}^{s-1-j}u_{yy} \\ &\quad -\sum_{i=0}^{s-1} \partial_{x}^{i}(\partial_{x} a_{2}\partial_{x}^{s-1-i}u_{y}) -\sum_{i=0}^{s-1}\partial_{x}^{i}(\partial_{x} a_{s-1-i}\partial_{x}^{s-1-i}u) \\ &\quad +\frac{s\partial_{x}a_{22}}{a_{22}}[a_{2}\partial_{x}^{s-1}u_{y} +a_{s-1}\partial_{x}^{s-1}u-f_{s-1}] :=\widetilde{f}_{s}. \end{align*} If $\varepsilon=\varepsilon(s)$ and $\theta$ are sufficiently small, we can apply Lemma \ref{lem4.1} to obtain \begin{equation} \label{e4.5} \|\partial_{x}^{s}u\|+\|\partial_{x}^{s}u_{y}\| \leq M\|\widetilde{f}_{s}\|. \end{equation} We now estimate each term of $\widetilde{f}_{s}$. Using Lemma \ref{lem4.2} (ii), Lemma \ref{lem2.2} (iii), and the fact that $\partial_{x}a_{22}$ vanishes outside of $X$, produces \begin{align*} \|\sum_{i=2}^{s-1}\sum_{j=2}^{i} \begin{pmatrix}i\\j\end{pmatrix} \partial_{x}^{j+1}a_{22}\partial_{x}^{s-1-j}u_{yy}\| &= \|\sum_{i=2}^{s-1}\sum_{j=2}^{i} \begin{pmatrix}i\\j\end{pmatrix} \partial_{x}^{j+1}a_{22}\partial_{x}^{s-1-j}u_{yy}\|_{0,X} \\ &\leq M_{1}(|\partial_{x}^{3}a_{22}|_{\infty}\| u\|_{s-1,X}+\|\partial_{x}^{3}a_{22}\|_{s-1,X}|u|_{\infty}) \\ & \leq M_{1}'(\| u\|_{s-1,X}+\| a_{22}\|_{s+2,X}|u|_{\infty}). \end{align*} A calculation shows that \[ \sum_{i=0}^{s-1}\partial_{x}^{i}(\partial_{x} a_{2}\partial_{x}^{s-1-i}u_{y})=s\partial_{x}a_{2}\partial_{x}^{s-1}u_{y} +\sum_{i=1}^{s-1}\sum_{j=1}^{i}\begin{pmatrix}i\\j\end{pmatrix} \partial_{x}^{j+1}a_{2}\partial_{x}^{s-1-j}u_{y}. \] Then using the same procedure as above, we have \[ \|\sum_{i=0}^{s-1}\partial_{x}^{i}(\partial_{x} a_{2}\partial_{x}^{s-1-i}u_{y})\|\leq M_{2}\| \partial_{x}^{s-1}u_{y}\|+M_{2}'(\| u\| _{s-1,X}+\| a_{2}\|_{s+2,X}|u|_{\infty}). \] Furthermore the following estimates are obtained in the same way: \begin{align*} &\|\sum_{i=0}^{s-1}\partial_{x}^{i}(\partial_{x} a_{s-1-i}\partial_{x}^{s-1-i}u)\| +\|\frac{s\partial_{x}a_{22}}{a_{22}}\sum_{i=0}^{s-2} \partial_{x}^{i}(\partial_{x} a_{s-2-i}\partial_{x}^{s-2-i}u)\| \\ &\leq M_{3}(\| u\|_{s-1,X}+(\|a\|_{s+2,X}+\| a_{1}\|_{s+2,X}+\| a_{11}\|_{s+2,X})|u|_{\infty}) \end{align*} and \[ \|\frac{s\partial_{x}a_{22}}{a_{22}}\sum_{i=0}^{s-2}\partial_{x}^{i}(\partial_{x} a_{2}\partial_{x}^{s-2-i}u_{y})\|\leq M_{4}(\| u\|_{s-1,X}+\| a_{2}\|_{s+2,X}|u|_{\infty}). \] Also since \[ \sum_{i=0}^{s-2}\partial_{x}^{i}(\partial_{x} a_{22}\partial_{x}^{s-2-i}u_{yy})=(s-1)\partial_{x}a_{22}\partial_{x}^{s-2}u_{yy} +\sum_{i=1}^{s-2}\sum_{j=1}^{i}\begin{pmatrix}i\\j\end{pmatrix} \partial_{x}^{j+1}a_{22}\partial_{x}^{s-2-j}u_{yy} \] and $\partial_{x}a_{22}=O(\varepsilon)$, we find that \begin{align*} &\|\frac{s\partial_{x}a_{22}}{a_{22}}\sum_{i=0}^{s-2} \partial_{x}^{i}(\partial_{x} a_{22}\partial_{x}^{s-2-i}u_{yy})\|\\ & \leq\varepsilon s^{2}M_{5}\|\partial_{x}^{s-2}u_{yy}\|_{0,X} +M_{5}'(\|u\|_{s-1,X}+\| a_{22}\|_{s+2,X}|u|_{\infty}), \end{align*} where $M_{5}$ is independent of $\varepsilon$ and $s$. Summing the above estimates produces \begin{equation} \label{e4.6} \|\widetilde{f}_{s}\|\leq M_{6}(\| f\|_{s} + \| u\|_{s-1,X} + \|\partial_{x}^{s-1}u_{y}\| +\varepsilon s^{2}\|\partial_{x}^{s-2}u_{yy}\|_{0,X} +\Lambda_{s+2} |u|_{\infty}). \end{equation} Therefore if we estimate $\|\partial_{x}^{s-2}u_{yy}\|_{0,X}$ appropriately and show that \[ |u|_{\infty}\leq M_{7}\| f\|_{2,X}, \] the proof will be complete by induction. We now estimate $\|\partial_{x}^{s-2}u_{yy}\|_{0,X}$. Differentiate the equation \[ \widetilde{L}u_{yy}:=-\theta u_{xxyy}+a_{22}u_{yy}=f-a_{11}u_{xx} -a_{1}u_{x}-a_{2}u_{y}-au:=\widetilde{g} \] with respect to $x$ $(s-2)$-times, then \[ \widetilde{L}\partial_{x}^{s-2}u_{yy}=\partial_{x}^{s-2}\widetilde{g} -\sum_{i=0}^{s-3}\partial_{x}^{i}(\partial_{x}a_{22}\partial_{x}^{s-3-i}u_{yy}) :=\widetilde{g}_{s-2}. \] Multiply the above equation by $\partial_{x}^{s-2}u_{yy}$ and integrate by parts in $X$ to obtain, \[ \|\partial_{x}^{s-2}u_{yy}\|_{0,X}\leq M_{8}\| \widetilde{g}_{s-2}\|_{0,X}. \] We now estimate $\|\widetilde{g}_{s-2}\|_{0,X}$. Using the same methods as above, we have \[ \|\partial_{x}^{s-2}(a_{1}u_{x}+a_{2}u_{y}+au) +\sum_{i=0}^{s-3}\partial_{x}^{i}(\partial_{x}a_{22}\partial_{x}^{s-3-i}u_{yy}) \|_{0,X}\leq M_{9}(\| u\|_{s-1,X}+\Lambda_{s+2}|u|_{\infty}). \] Furthermore \[ \partial_{x}^{s-2}(a_{11}u_{xx})=a_{11}\partial_{x}^{s}u +\sum_{i=1}^{s-2}\begin{pmatrix}s-2\\i\end{pmatrix} \partial_{x}^{i}a_{11}\partial_{x}^{s-2-i}u_{xx}; \] thus \[ \|\partial_{x}^{s-2}(a_{11}u_{xx})\|_{0,X}\leq M_{10}(\|\partial_{x}^{s}u\|_{0,X}+\| u\|_{s-1,X}+\Lambda_{s+2}|u|_{\infty}). \] It follows that \begin{equation} \label{e4.7} \|\partial_{x}^{s-2}u_{yy}\|_{0,X} \leq M_{11}(\|\partial_{x}^{s}u\|_{0,X} +\| u\|_{s-1,X}+\Lambda_{s+2}|u|_{\infty}). \end{equation} The coefficient of $\|\partial_{x}^{s-2}u_{yy}\|_{0,X}$ in \eqref{e4.6} is $\varepsilon s^{2}M_{6}$. If $\varepsilon=\varepsilon(s)$ is chosen sufficiently small so that $\varepsilon s^{2}MM_{6}M_{11}<\frac{1}{2}$, we can then bring $\varepsilon s^{2}MM_{6}M_{11}\|\partial_{x}^{s}u\|_{0,X}$ from \eqref{e4.7} to the left-hand side of \eqref{e4.5}, so that by induction on $s$ \[ \|\partial_{x}^{s}u\|+\|\partial_{x}^{s}u_{y}\| \leq M_{6}'(\| f\|_{s}+\| u\|_{s-1,X}+\Lambda_{s+2}(|u|_{\infty}+\| f\|_{2,X})). \] We now estimate $|u|_{\infty}$ to complete the proof. The above methods can be used to show that \[ \| u\|_{2,X}\leq M_{12}\| f\|_{2,X}. \] Then by the Sobolev lemma, \[ |u|_{\infty}\leq M_{13}\| u\|_{2,X}\leq M_{13}'\| f\|_{2,X}. \] \end{proof} We now estimate the remaining derivatives. \begin{proposition} \label{prop4.2} Let $u$, $f$, $s$, and $\varepsilon$ be as in Proposition \ref{prop4.1}. Then \[ \|\partial_{x}^{\alpha}\partial_{y}^{\beta}u\|_{0,X} \leq C_{s}(\| f\|_{s,X}+\| u\|_{s-1,X} +\Lambda_{s+2}\| f\|_{2,X}) \] for $\alpha+\beta\leq s$, where $C_{s}$ is independent of $\varepsilon$ and $\theta$. \end{proposition} \begin{proof} The cases $\beta=0,1,2$ follow from \eqref{e4.7} and Proposition \ref{prop4.1}. We proceed by induction on $\beta$. Assume that the desired estimate holds for $0\leq\alpha\leq s-\beta$, and $0\leq\beta\leq k-1$, for some $k\leq s$. Differentiate the equation \[ \widetilde{L}u_{yy}:=-\theta u_{xxyy}+a_{22}u_{yy}=f-a_{11}u_{xx} -a_{1}u_{x}-a_{2}u_{y}-au:=\widetilde{g} \] with respect to $\partial_{x}^{\alpha}\partial_{y}^{k-2}$ where $0\leq\alpha\leq s-k$, then \begin{align*} \widetilde{L}\partial_{x}^{\alpha}\partial_{y}^{k}u &= \partial_{x}^{\alpha}\partial_{y}^{k-2}\widetilde{g} -\sum_{i=0}^{\alpha-1}\partial_{y}^{k-2}\partial_{x}^{i}(\partial_{x} a_{22}\partial_{x}^{\alpha-1-i}u_{yy})-\sum_{i=0}^{k-3}\partial_{y}^{i} (\partial_{y}a_{22}\partial_{y}^{k-3-i}\partial_{x}^{\alpha}u_{yy})\\ &:=\widetilde{g}_{\alpha,k-2}. \end{align*} Multiply the above equation by $\partial_{x}^{\alpha}\partial_{y}^{k}u$, and integrate by parts in X to obtain \[ \|\partial_{x}^{\alpha}\partial_{y}^{k}u\|_{0,X}\leq M\|\widetilde{g}_{\alpha,k-2}\|_{0,X}. \] We now estimate $\|\widetilde{g}_{\alpha,k-2}\|_{0,X}$. Using Lemma \ref{lem4.2} (ii), we have \begin{align*} &\|\partial_{x}^{\alpha}\partial_{y}^{k-2}(a_{11}u_{xx}) \|_{0,X}\\ &\leq M_{1}(\|\partial_{x}^{\alpha+2}\partial_{y}^{k-2}u\|_{0,X}+ \sum_{p\leq\alpha, q\leq k-2,\; (p,q)\neq(0,0)} \|\partial_{x}^{p}\partial_{y}^{q}a_{11} \partial_{x}^{\alpha-p}\partial_{y}^{k-2-q}u_{xx}\|_{0,X})\\ &\leq M_{1}'(\|\partial_{x}^{\alpha+2}\partial_{y}^{k-2}u\|_{0,X} +|a_{11}|_{C^{1}(\overline{X})}\| u \|_{s-1,X}+\|a_{11} \|_{s,X} |u|_{\infty}) \\ &\leq M_{1}''(\|\partial_{x}^{\alpha+2}\partial_{y}^{k-2}u\|_{0,X} +\| u\|_{s-1,X}+\Lambda_{s+2}\| f\|_{2,X}). \end{align*} Furthermore if $\alpha5$. Define a sequence of domains $\{X_{n}\}_{n=1}^{\infty}$ by \[ X_{1}=X, \quad X_{n}=(1-\sum_{i=1}^{n-1}\mu^{-i})X, \] where $\lambda X=\{\lambda x: x\in X\}$. Then $X_{\infty}=(1-\frac{1}{\mu-1})X$. In addition, let $\frac{3}{2}<\tau<2$ and define $\mu_{n}=\mu^{\tau^{n+n_{0}}}$, where $n_{0}>0$ will be chosen sufficiently large. We now construct smoothing operators on $L^{2}(X_{n})$. Fix $\widehat{\psi}\in C^{\infty}_{c}(\mathbb{R}^{2})$ such that $\widehat{\psi}\equiv 1$ in some neighborhood of the origin. Let $\psi(x)=\iint_{\mathbb{R}^{2}}\widehat{\psi}(\eta)e^{2\pi i\eta\bullet x}d\eta$ be the inverse Fourier transform of $\widehat{\psi}$. Then $\psi$ is a Schwartz function and satisfies $\iint_{\mathbb{R}^{2}}\psi(x)dx=1$, and $\iint_{\mathbb{R}^{2}}x^{\alpha}\psi(x)dx=0$ for any multi-index $\alpha\neq 0$. If $g\in L^{2}(\mathbb{R}^{2})$ and $\gamma\geq 1$, we define the smoothing operators $S_{\gamma}':L^{2}(\mathbb{R}^{2})\to H^{\infty}(\mathbb{R}^{2})$ by \[ (S_{\gamma}'g)(x)=\gamma^{2}\iint_{\mathbb{R}^{2}}\psi(\gamma(x-y))g(y)dy. \] Then we have the following result (see \cite{s1}). \begin{lemma} \label{lem5.1} Let $a,b\in\mathbb{Z}_{\geq0}$ and $g\in H^{a}(\mathbb{R}^{2})$, then \begin{itemize} \item[(i)] $\|S_{\gamma}'g\|_{H^{b}(\mathbb{R}^{2})}\leq C_{a,b}\| g\|_{H^{a}(\mathbb{R}^{2})}$, $b\leq a$, \item[(ii)] $\|S_{\gamma}'g\|_{H^{b}(\mathbb{R}^{2})}\leq C_{a,b}\gamma^{b-a}\| g\|_{H^{a}(\mathbb{R}^{2})}$, $a\leq b$, \item[(iii)] $\| g-S_{\gamma}'g\|_{H^{b}(\mathbb{R}^{2})}\leq C_{a,b}\gamma^{b-a}\| g\|_{H^{a}(\mathbb{R}^{2})}$, $b\leq a$. \end{itemize} \end{lemma} To complete the construction, we also need the following extension theorem. \begin{theorem}[\cite{s2}] \label{thm5.1} Let $D$ be a bounded convex domain in $\mathbb{R}^{2}$ with Lipschitz smooth boundary. Then there exists a linear operator $T_{D}:L^{2}(D)\to L^{2}(\mathbb{R}^{2})$ such that: \begin{itemize} \item[(i)] $T_{D}(g)|_{D}=g$, \item[(ii)] $T_{D}:H^{a}(D)\to H^{a}(\mathbb{R}^{2})$ continuously for each $a\in\mathbb{Z}_{\geq 0}$. \end{itemize} \end{theorem} To obtain smoothing operators on $X_{n}$, $S_{n}:L^{2}(X_{n})\to H^{\infty}(X_{n})$, we set $S_{n}g=(S'_{\mu_{n}}T_{X_{n}}g)|_{X_{n}}$. Furthermore, it is clear that the corresponding results of Lemma \ref{lem5.1} hold for each $S_{n}$. We now set up the iteration procedure. A sequence of functions $\{w_{n}\}_{n=1}^{\infty}$ will be shown to converge to a solution of \eqref{e5.1}, and shall be defined inductively as follows. Set $w_{1}=0$ and suppose that $w_{j}$, $j\leq n$, are already defined in $X_{j}$, then set $w_{n+1}=w_{n}+S_{n}u_{n}$ in $X_{n+1}$, where $u_{n}$ is defined in $X_{n}$ and will be specified below. Set $f_{n}=-\Phi(w_{n})$ in $X_{n}$, and let $\phi_{n}$ be a $C^{\infty}$ cut-off function \[ \phi_{n}(x)=\begin{cases} 1 & \text{if $x\in X_{n+1}$},\\ 0 & \text{if $x\in X-X_{n}$,} \end{cases} \] such that \[ |\phi_{n}|_{C^{s}(X_{n})}\leq M_{s}\mu^{sn}. \] Let \[ L(w_{n})=\sum_{i,j}a_{ij}(w_{n})\partial_{ij}+\sum_{i}a_{i}(w_{n})\partial_{i} +a(w_{n}) \] denote the linearization of $\Phi(w)$ evaluated at $w_{n}$, and let $\{\theta_{n}\}_{n=1}^{\infty}$ be a sequence of positive numbers tending towards zero that will be specified later. Then define $u_{n}$ in $X_{n}$ by $u_{n}=v_{n}|_{X_{n}}$, where $v_{n}$ is the solution of \[ L_{\theta_{n}}(w_{n})v_{n}=\phi_{n}f_{n} \quad \text{in } X, \] given by Theorem \ref{thm2.1}. Since $\mu>5$ we have $\frac{3}{4}X\subset X_{\infty}$. Therefore, it follows from the definition of $\Phi(w)$ in \eqref{e1.5} that the coefficients of $L_{\theta_{n}}(w_{n})$ are well-defined in all of $X$, even though $w_{n}$ is only defined in $X_{n}$. For simplicity we denote the Sobolev norms $\|\cdot\|_{H^{s}(X_{n})}$ by $\|\cdot\|_{s}^{n}$, and the $C^{s}(\overline{X}_{n})$ norms by $|\cdot|_{s}^{n}$. Let $s_{*}\in\mathbb{Z}_{\geq 0}$ be fixed such that $\Phi(0)\in H^{s_{*}}(X)$, and define \[ \sigma=n(n+1)\tau^{-(n+1+n_{0})}, \quad \delta=\frac{16}{\tau-1}. \] The convergence of the sequence $\{w_{n}\}_{n=1}^{\infty}$ to a solution of \eqref{e5.1} will follow from the following four statements. Each will be proven by induction on $j$, for some constants $C_{1}$, $C_{2}$, and $C_{3}$ independent of $j$ and dependent on $\mu$ and $s_{*}$. We shall require that $s\leq s_{*}-18-2\delta-\frac{6\tau}{2-\tau}$ and $s_{*}\geq 22+2\delta +\frac{6\tau}{2-\tau}$. \begin{itemize} \item[(I$_{j}$)] $\| w_{j}\|_{s+15}^{j}\leq\mu_{j}^{\sigma s+\delta}\| f_{1}\|^{1}_{s_{*}-15}$ \item[(II$_{j}$)] $\| u_{j-1}\|_{s}^{j-1}\leq C_{1}\mu_{j-1} ^{\tau^{-1}(s-s_{*}+18+2\delta)}\| f_{1}\|^{1}_{s_{*}-15}$ \item[(III$_{j}$)] $\| f_{j}\|_{s}^{j}\leq C_{2}\mu_{j} ^{\tau^{-1}(s-s_{*}+18+2\delta)}\| f_{1}\|^{1}_{s_{*}-15}$ \item[(IV$_{j}$)] $\| w_{j}\|^{j}_{14}\leq C_{3}$ \end{itemize} To start the induction process observe that I$_{1}$, II$_{1}$, and IV$_{1}$ are trivial, and that III$_{1}$ holds if we set $C_{2}=\mu_{1}$. Now assume that I$_{j}$,$\dots$,IV$_{j}$ hold for $1\leq j\leq n$. The next four propositions will prove the induction step. Note that the coefficients of $L(w_{j})$ satisfy the conditions placed on \eqref{e2.1} with $r=s_{*}-2$. Therefore the results of the previous sections apply to $L_{\theta_{j}}(w_{j})$, $1\leq j\leq n$, as long as $\varepsilon(s_{*})$ and $\theta_{j}$ are sufficiently small and $s\leq s_{*}-15$. \begin{proposition} \label{prop5.1} If $s\leq s_{*}-15$ and $\mu(s_{*})$ is sufficiently large, then \[ \| w_{n+1}\|_{s+15}^{n+1}\leq\mu_{n+1}^{\sigma s+\delta}\| f_{1}\|^{1}_{s_{*}-15}. \] \end{proposition} \begin{proof} We have \[ \| w_{n+1}\|^{n+1}_{s+15}\leq \| w_{n}\|_{s+15}^{n}+\| S_{n}u_{n}\| _{s+15}^{n}. \] Furthermore by Theorem \ref{thm4.2} and Lemma \ref{lem4.2} (iii), \begin{align*} \| S_{n}u_{n}\|_{s+15}^{n} &\leq M_{1}\mu_{n}^{15}\| u_{n}\|_{s}^{n}\\ &\leq M_{2}\mu_{n}^{15}(\|\phi_{n}f_{n}\|_{s}^{n}+ \|w_{n}\|_{s+15}^{n}\|\phi_{n}f_{n}\|_{2}^{n}). \end{align*} Using Lemma \ref{lem4.2} (ii), we obtain \begin{align*} \|\phi_{n}f_{n}\|_{s}^{n} &\leq M_{3}(\|f_{n}\|_{s}^{n}+\|\phi_{n}\|_{s}^{n} |f_{n}|_{0}^{n})\\ &\leq M_{4}(\|f_{n}\|_{s}^{n}+\|\phi_{n}\|_{s}^{n} \| f_{n}\|_{2}^{n})\\ &\leq M_{5}\mu^{sn}\| f_{n}\|_{s}^{n}. \end{align*} Moreover by the definition of $f_{n}$ and Lemma \ref{lem4.2} (iii) \begin{equation} \label{e5.2} \| f_{n}\|_{s}^{n}\leq M_{6}(\| f_{1}\|_{s_{*}-15}^{n}+\| w_{n}\|_{s+4}^{n}), \end{equation} so that \[ \|\phi_{n}f_{n}\|_{s}^{n}\leq M_{7}\mu^{sn} (\| f_{1}\|_{s_{*}-15}^{n}+\| w_{n}\|_{s+4}^{n}). \] Similarly using IV$_{n}$, \[ \|\phi_{n}f_{n}\|_{2}^{n}\leq M_{7}\mu^{2n} (\| f_{1}\|_{s_{*}-15}^{n}+\| w_{n}\|_{6}^{n})\leq M_{8}\mu^{2n}. \] We now have \[ \| S_{n}u_{n}\|_{s+15}^{n}\leq M_{9}\mu_{n}^{16} \mu^{sn}(\| f_{1}\|_{s_{*}-15}^{1}+\| w_{n}\|_{s+15}^{n}). \] Therefore \begin{align*} \| w_{n+1}\|_{s+15}^{n+1}&\leq 2M_{9} \mu_{n}^{16}\mu^{sn}(\| f_{1}\|_{s_{*}-15}^{1}+ \| w_{n}\|_{s+15}^{n})\\ &\leq \mu_{n}^{16}\mu^{2sn}(\| f_{1}\|_{s_{*}-15}^{1}+\| w_{n}\|_{s+15}^{n}), \end{align*} where the last inequality holds if $\mu$ is chosen so large that $2M_{9}\mu^{-1}\leq 1$. It follows that \[ \| w_{n+1}\|_{s+15}^{n+1}\leq (\prod_{i=1}^{n} \mu_{i}^{16}\mu^{2si})M_{10}\|f_{1}\|_{s_{*}-15}^{1}, \] where \[ M_{10}=1+\mu_{1}^{-16}\mu^{-2s}+\dots+\prod_{i=1}^{n-1}\mu_{i}^{-16}\mu^{-2si} \leq 2, \] if $\mu$ is large. Hence \[ \| w_{n+1}\|_{s+15}^{n+1} \leq 2\mu^{sn(n+1)+\frac{16}{\tau-1}(\tau^{n+1+n_{0}}-\tau^{1+n_{0}})}\| f_{1} \|_{s_{*}-15}^{1} \leq \mu_{n+1}^{\sigma s+\delta}\| f_{1} \|_{s_{*}-15}^{1}, \] where $\sigma=n(n+1)\tau^{-(n+1+n_{0})}$ and $\delta=\frac{16}{\tau-1}$. \end{proof} \begin{proposition} \label{prop5.2} If $s\leq s_{*}-20-2\delta$ and $n_{0}(s_{*})$ is sufficiently large then \[ \| u_{n}\|_{s}^{n}\leq C_{1}\mu_{n}^{\tau^{-1}(s-s_{*}+18+2\delta)} \| f_{1}\|^{1}_{s_{*}-15}, \] where $C_{1}$ depends on $\mu$ and $s_{*}$. \end{proposition} \begin{proof} By Theorem \ref{thm4.2} \[ \| u_{n}\|_{s_{*}-15}^{n}\leq M_{1} (\|\phi_{n}f_{n}\|_{s_{*}-15}^{n}+\| w_{n} \|_{s_{*}}^{n}\|\phi_{n}f_{n}\|_{2}^{n}), \] where $M_{1}$ depends only on $s_{*}$. By Lemma \ref{lem4.2} (ii), \eqref{e5.2}, and I$_{n}$ \begin{align*} \|\phi_{n}f_{n}\|_{s_{*}-15}^{n}&\leq M_{2}( \|f_{n}\|_{s_{*}-15}^{n}+\|\phi_{n}\|_{s_{*}-15}^{n} \| f_{n}\|_{2}^{n})\\ &\leq M_{3}(1+\mu^{(s_{*}-15)n})\mu_{n}^{\sigma(s_{*}-26)+\delta}\| f_{1}\|_{s_{*}-15}^{1}\\ &\leq M_{4}\mu_{n}^{2s_{*}\sigma+\delta}\| f_{1}\|_{s_{*}-15}^{1}, \end{align*} where $M_{3}$ depends only on $s_{*}$. Similarly III$_{n}$ yields \[ \|\phi_{n}f_{n}\|^{n}_{2}\leq M_{5}C_{2}\mu_{n}^{2\sigma+\tau^{-1}(20-s_{*}+2\delta)}\| f_{1}\|_{s_{*}-15}^{1}. \] Therefore for some constant $M_{6}$ depending on $\mu$ and $s_{*}$, we have \begin{equation} \label{e5.3} \begin{aligned} \| u_{n}\|_{s_{*}-15}^{n} &\leq M_{6}(\mu_{n}^{2s_{*}\sigma+\delta}+\mu_{n}^{\sigma(s_{*}-15) +\delta}\mu_{n}^{ 2\sigma+\tau^{-1}(20-s_{*}+2\delta)}) \| f_{1}\|_{s_{*}-15}^{1}\\ &\leq 2M_{6}\mu_{n}^{2s_{*}\sigma+\delta}\| f_{1}\|_{s_{*}-15}^{1}, \end{aligned} \end{equation} since $s_{*}\geq 20+2\delta$. Furthermore Lemma \ref{lem2.3} and III$_{n}$ produce \[ \| u_{n}\|^{n}_{0}\leq M_{7}\| f_{n}\|^{n}_{0}\leq M_{7}C_{2}\mu_{n}^{\tau^{-1}(18-s_{*}+2\delta)}\| f_{1}\|_{s_{*}-15}^{1}. \] Then applying Lemma \ref{lem4.2} (i), we find \begin{align*} \| u_{n}\|^{n}_{s} &\leq M_{8}(\|u_{n}\|^{n}_{0})^{1-\frac{s}{s_{*}-15}}(\| u_{n}\|^{n}_{s_{*}-15})^{\frac{s}{s_{*}-15}}\\ &\leq M_{9}\mu_{n}^{\tau^{-1}(18-s_{*}+2\delta) (1-\frac{s}{s_{*}-15})+(2s_{*}\sigma+\delta)(\frac{s}{s_{*}-15})}\| f_{1}\|_{s_{*}-15}^{1}\\ &\leq M_{9}\mu_{n}^{\tau^{-1}(s-s_{*}+18+2\delta)}\| f_{1}\|_{s_{*}-15}^{1} \end{align*} if $\sigma$ is sufficiently small. Note that $\sigma$ may be made arbitrarily small by choosing $n_{0}$ sufficiently large. We then set $C_{2}=M_{9}$ to obtain the desired result. \end{proof} \begin{proposition} \label{prop5.3} If $s\leq s_{*}-18-2\delta-\frac{6\tau}{2-\tau}$, $s_{*}\geq 22+2\delta +\frac{6\tau}{2-\tau}$, $n_{0}(s_{*})$ and $\mu(s_{*})$ are sufficiently large, and $\varepsilon(s_{*})$ is sufficiently small, then \[ \| f_{n+1}\|_{s}^{n+1}\leq C_{2}\mu_{n+1} ^{\tau^{-1}(s-s_{*}+18+2\delta)}\| f_{1}\|^{1}_{s_{*}-15}. \] \end{proposition} \begin{proof} Expanding $\Phi(w_{n+1})$ in a Taylor series yields \[ f_{n+1}=f_{n}-L(w_{n})S_{n}u_{n}+Q_{n}=f_{n}-\theta_{n} (S_{n}u_{n})_{\eta\eta\xi\xi}-L_{\theta_{n}}(w_{n})S_{n}u_{n}+Q_{n}, \] where $(\xi,\eta)$ are the change of variables given in section \S 2 by \[ a_{12}(w_{n})\xi_{x}+a_{22}(w_{n})\xi_{y}=0 \quad \text{in } X, \quad \xi(x,0)=x, \quad \xi(\pm x_{0},y)=\pm x_{0}, \quad \eta=y, \] and where $Q_{n}$ is the quadratic error term given by \[ Q_{n}=\int_{0}^{1}(t-1)\partial_{t}^{2}\Phi(w_{n}+tS_{n}u_{n})dt. \] Since $L_{\theta_{n}}(w_{n})u_{n}=f_{n}$ in $X_{n+1}$ we have \begin{equation} \label{e5.4} f_{n+1}=L_{\theta_{n}}(w_{n})(u_{n}-S_{n}u_{n}) -\theta_{n}(S_{n}u_{n})_{\eta\eta\xi\xi}+Q_{n}, \end{equation} in $X_{n+1}$. Each term of \eqref{e5.4} shall be estimated separately. First note that $\theta_{n}$ may be chosen sufficiently small to guarantee that \[ \|\theta_{n}(S_{n}u_{n})_{\eta\eta\xi\xi}\|_{s}^{n+1} \leq\frac{1}{3}C_{2}\mu_{n+1}^{\tau^{-1}(s-s_{*}+18+2\delta)} \| f_{1}\|_{s_{*}-15}^{1}. \] We now estimate $L_{\theta_{n}}(w_{n})(u_{n}-S_{n}u_{n})$. By Lemma \ref{lem4.2} and IV$_{n}$, \begin{align*} &\|L_{\theta_{n}}(w_{n})(u_{n}-S_{n}u_{n})\|_{s}^{n+1}\\ &\leq \|L_{\theta_{n}}(w_{n})(u_{n}-S_{n}u_{n})\|_{s}^{n}\\ &\leq M_{1}(\| u_{n}-S_{n}u_{n}\|_{s+2}^{n}+ \| w_{n}\|_{s+4}^{n}|u_{n}-S_{n}u_{n}|_{0}^{n}) +O(\theta_{n})\\ &\leq M_{2}(\| u_{n}-S_{n}u_{n}\|_{s+2}^{n}+ \| w_{n}\|_{s+4}^{n}\| u_{n}-S_{n}u_{n}\|_{2}^{n}) +O(\theta_{n})\\ &\leq M_{3}(\mu_{n}^{s+17-s_{*}}\| u_{n}\|_{s_{*}-15}^{n}+\mu_{n}^{17-s_{*}}\| w_{n}\|_{s+4}^{n}\| u_{n}\|_{s_{*}-15}^{n}) +O(\theta_{n}). \end{align*} Furthermore by \eqref{e5.3}, \[ \| u_{n}\|_{s_{*}-15}^{n}\leq M_{4}\mu_{n}^{2s_{*}\sigma+\delta}\| f_{1}\|_{s_{*}-15}^{1}. \] If $\theta_{n}$ and $\sigma$ are sufficiently small and $\mu$ is sufficiently large, it follows that \begin{align*} \| L_{\theta_{n}}(w_{n})(u_{n}-S_{n}u_{n})\|_{s}^{n+1} &\leq M_{5}\mu^{-1}(\mu_{n}^{3s_{*}\sigma+s-s_{*}+17+\delta}+ \mu_{n}^{3s_{*}\sigma-s_{*}+17+2\delta})\| f_{1}\|_{s_{*}-15}^{1}\\ &\leq \frac{1}{3}C_{2}\mu_{n+1}^{\tau^{-1}(s-s_{*}+18+2\delta)} \| f_{1}\|_{s_{*}-15}^{1}. \end{align*} We now estimate $Q_{n}$. Apply Lemma \ref{lem4.2} (ii) to obtain \begin{align*} \| Q_{n}\|_{s}^{n+1} &\leq \|Q_{n}\|_{s}^{n}\\ &\leq \int_{0}^{1}\sum_{|\alpha|,|\beta|,|\rho|\leq 2}\| \partial^{\rho}\Phi(w_{n}+tS_{n}u_{n})\partial^{\alpha}(S_{n}u_{n}) \partial^{\beta}(S_{n}u_{n})\|_{s}^{n}dt\\ &\leq \int_{0}^{1}\sum_{|\alpha|,|\beta|,|\rho|\leq 2}M_{6}(| \partial^{\rho}\Phi(w_{n}+tS_{n}u_{n})|_{0}^{n}\|\partial^{\alpha}(S_{n}u_{n}) \partial^{\beta}(S_{n}u_{n})\|_{s}^{n}\\ &\quad +\|\partial^{\gamma}\Phi(w_{n}+tS_{n}u_{n})\|_{s}^{n} |\partial^{\alpha}(S_{n}u_{n}) \partial^{\beta}(S_{n}u_{n})|_{0}^{n})dt. \end{align*} Then the Sobolev lemma and the interpolation inequality $\|u^{2}\|_{L^{2}}\leq C\| u\|_{H^{1}}^{2}$, show that \begin{align*} \| Q_{n}\|_{s}^{n+1} &\leq \int_{0}^{1}\sum_{|\rho|\leq 2}M_{7}(\| \partial^{\rho}\Phi(w_{n}+tS_{n}u_{n})\|_{2}^{n} (\| S_{n}u_{n}\|_{s+3}^{n})^{2}\\ & \quad +\|\partial^{\rho}\Phi(w_{n}+tS_{n}u_{n})\|_{s}^{n} (\| S_{n}u_{n}\|_{4}^{n})^{2})dt. \end{align*} Furthermore by Lemma \ref{lem4.2} (iii), I$_{n}$, IV$_{n}$, and Proposition \ref{prop5.2} \begin{align*} \| Q_{n} \|_{s}^{n+1} &\leq M_{8} [(\| w_{n} \|_{6}^{n} +\mu_{n}^{2}\| u_{n} \|_{4}^{n})(\mu_{n}^{3}\|u_{n} \|_{s}^{n})^{2} + (\|w_{n} \|_{s+4}^{n} +\mu_{n}^{4}\|u_{n} \|_{s}^{n})(\| u_{n} \|_{4}^{n})^{2}] \\ &\leq (M_{9}\| f_{1}\|_{s_{*}-15}^{1}) [(1+\mu_{n}^{2+\tau^{-1}(-s_{*}+22+ 2\delta)})\mu_{n}^{6+2\tau^{-1}(s-s_{*}+18+2\delta)} \\ &\quad +(\mu_{n}^{\sigma(s-11)+\delta}+\mu_{n}^{4+\tau^{-1}(s-s_{*}+18 +2\delta)})\mu_{n}^{2\tau^{-1}(-s_{*}+22+2\delta)}] \| f_{1}\|_{s_{*}-15}^{1} \\ &\leq (M_{10}\| f_{1}\|_{s_{*}-15}^{1})\mu_{n}^{s-s_{*}+18+2\delta}\| f_{1}\|_{s_{*}-15}^{1}, \end{align*} since $s\leq s_{*}-18-2\delta-\frac{6\tau}{2-\tau}$ and $s_{*}\geq 22+2\delta+\frac{6\tau}{2-\tau}$. If $\varepsilon(s_{*})$ is sufficiently small to guarantee that $M_{10}\|f_{1}\|_{s_{*}-15}^{1}\leq\frac{1}{3}C_{2}$, then \[ \| Q_{n}\|_{s}^{n+1}\leq \frac{1}{3}C_{2}\mu_{n+1}^{\tau^{-1}(s-s_{*}+18+2\delta)} \| f_{1}\|_{s_{*}-15}^{1}. \] By combining the estimates for each term of \eqref{e5.4} we obtain the desired result. \end{proof} \begin{proposition} \label{prop5.4} If $n_{0}(s_{*})$ is sufficiently large then \[ \| w_{n+1}\|_{14}^{n+1}\leq C_{3}, \] where $C_{3}$ depends on $\mu$ and $s_{*}$. \end{proposition} \begin{proof} Let $a=14+\tau^{-1}(18+2\delta-s_{*})$ and note that since $s_{*}\geq 22+2\delta+\frac{6\tau}{2-\tau}$, $\tau\geq\frac{3}{2}$, we have $a<0$. If $n_{0}$ is sufficiently large, we may apply Proposition \ref{prop5.2} to obtain \[ \| w_{n+1}\|_{14}^{n+1} \leq \sum_{i=1}^{n}\| S_{i}u_{i}\|_{14}^{i} \leq \sum_{i=1}^{n}\mu_{i}^{14}\| u_{i}\|_{0}^{i} \leq \sum_{i=1}^{\infty}\mu_{i}^{a}\| f_{1}\|_{s_{*}-15}^{1}:=C_{3}. \] \end{proof} To obtain the largest value for $s$ and smallest lower bound for $s_{*}$ which satisfy the conditions of Propositions \ref{prop5.1}, \ref{prop5.2}, \ref{prop5.3}, \ref{prop5.4}, we choose $\tau=1.6$ so that $s_{*}\geq 100$ and $s\leq s_{*}-96$. We now establish two corollaries which will complete the proof of Theorem \ref{thm1.3}. \begin{corollary} \label{coro5.1} $w_{n}\to w$ in $H^{s_{*}-96}(X_{\infty})$. \end{corollary} \begin{proof} If $s\leq s_{*}-96$ then by II$_{n}$, \[ \| w_{i}-w_{j}\|_{s}^{\infty} \leq \sum_{k=j}^{i}\| u_{k}\|_{s}^{k} \leq C_{1}\sum_{k=j}^{i}\mu_{k}^{\tau^{-1}(s-s_{*}+18+2\delta)}\| f_{1}\|_{s_{*}-15}^{1}. \] Hence $\{w_{n}\}$ is Cauchy in $H^{s}(X_{\infty})$ for all $s\leq s_{*}-96$, since $18+2\delta<96$. \end{proof} \begin{corollary} \label{coro5.2} $\Phi(w_{n})\to 0$ in $H^{s_{*}-96}(X_{\infty})$. \end{corollary} \begin{proof} If $s\leq s_{*}-96$ then by III$_{n}$, \[ \| \Phi(w_{n})\|_{s}^{\infty} \leq \| f_{n} \|_{s}^{n} \leq C_{2}\mu_{n}^{\tau^{-1}(s-s_{*}+18+2\delta)}\| f_{1}\|_{s_{*}-15}^{1}\to 0. \] \end{proof} Since $s_{*}\geq 100$, it follows that $w_{n}\to w$ in $C^{2}(\overline{X}_{\infty})$. Therefore $\Phi(w_{n})\to\Phi(w)$, showing that $w$ is a solution of \eqref{e5.1}. 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