\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 24, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/24\hfil Dirichlet problems] {Growth rate and existence of solutions to Dirichlet problems for prescribed mean curvature equations on unbounded domains} \author[Z. Jin\hfil EJDE-2008/24\hfilneg] {Zhiren Jin} \address{Zhiren Jin \newline Department of Mathematics and Statistics \\ Wichita State University \\ Wichita, Kansas 67260-0033, USA} \email{zhiren@math.wichita.edu} \thanks{Submitted February 9, 2008. Published February 22, 2008.} \subjclass[2000]{35J25, 35J60, 35J65} \keywords{Elliptic boundary-value problem; quasilinear elliptic equation; \hfill\break\indent prescribed mean curvature equation; unbounded domain; Perron's method} \begin{abstract} We prove growth rate estimates and existence of solutions to Dirichlet problems for prescribed mean curvature equation on unbounded domains inside the complement of a cone or a parabola like region in $\mathbb{R}^n$ ($n\geq 2$). The existence results are proved using a modified Perron's method by which a subsolution is a solution to the minimal surface equation, while the role played by a supersolution is replaced by estimates on the uniform $C^{0}$ bounds on the liftings of subfunctions on compact sets. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction and main results} Let $\Omega$ be an unbounded domain with $C^{2,\gamma }$ ($0<\gamma <1$) boundary in $\mathbb{R}^n$ ($n\geq 2$), $\phi$ be a $C^{0}$ function on $\partial \Omega$, and $\Lambda$ be a $C^{1}$ function on $\overline{\Omega}$, we consider the Dirichlet problem for the prescribed mean curvature equation on $\Omega$ (here the summation convention is used): \begin{gather} ((1+|Du|^{2}) \delta_{ij} - D_{i}u D_{j}u )D_{ij} u = n \Lambda (1+|Du|^{2})^{3/2} \quad\text{on } \Omega; \label{eq:problem11} \\ u=\phi \quad\text{on } \partial\Omega . \label{eq:problem12} \end{gather} In this paper, we investigate the conditions from which we can derive growth estimates and existence of solutions $u$ for \eqref{eq:problem11}-\eqref{eq:problem12}. When $\Omega$ is a bounded domain, Serrin proved in \cite{Serrin} that \eqref{eq:problem11}-\eqref{eq:problem12} has a solution in $C^{0}(\overline{\Omega})\cap C^{2}(\Omega )$ as long as one can get $C^{0}$ estimates and the mean curvature $H'$ on the boundary $\partial \Omega$ with respect to the inner normal satisfying $H'\geq \frac{n}{n-1} |\Lambda |$ on $\partial \Omega$. Furthermore, a counterexample is given \cite[page 480]{Serrin} to show that for some functions $\Lambda$, \eqref{eq:problem11}-\eqref{eq:problem12} do not have a $C^{2}$ solution (the only thing that did not work out in the example is the $C^{0}$ estimate). When $n=2$, $\Omega$ is a strip and $\Lambda$ is a constant $H$, there have been a lot of interest in investigating the solutions of \eqref{eq:problem11}-\eqref{eq:problem12}. Finn \cite{Finn1} showed that the solvability of \eqref{eq:problem11} in $\Omega$ implies that the width of $\Omega$ will be less than $\frac{1}{|H|}$. When the width of a strip $\Omega$ is $1/|H|$, the half cylinder of radius $1/(2|H|)$ is a graph with constant mean curvature $H$ in the strip. Collin \cite{Collin} and Wang \cite{Wang} showed independently that there are graphs with constant mean curvature $H$ on the strip $\Omega$ with width $1/|H|$ other than the half cylinder. When $\Lambda =H$ and $\Omega$ is an unbounded convex domain on a plane, Lopez \cite{Lopez2} proved that the necessary and sufficient condition for \eqref{eq:problem11} to have solutions with zero boundary value is that $\Omega$ is inside a strip of width $1/|H|$. When $\Omega$ is a strip on the plane, the existence of constant mean curvature graphs with prescribed boundary was considered by Lopez in \cite{Lopez4}. The approach used in \cite{Lopez4} is a modified version of the classical Perron's method of super- sub- solutions. The subsolution used in \cite{Lopez4} is a solution to the minimal surface equation (i.e. a solution to \eqref{eq:problem11}-\eqref{eq:problem12} with $\Lambda =0$), while the role played by a supersolution is replaced by a family of turned to side nodoids (the use of turned to side nodiods was adopted from an idea used by Finn \cite{Finn}) that were used to prove that liftings from subfunctions will be bounded uniformly on any compact subset of $\Omega$. When $\Omega$ is an unbounded domain inside a cone or cylinder, we proved in \cite{Jin3} the existence of solutions to \eqref{eq:problem11}-\eqref{eq:problem12} for certain class of functions $\Lambda$. The approach used in \cite{Jin3} is also a modified version of the classical Perron's method. There are new difficulties in carrying out the Perron's method when $\Lambda$ is not a constant and $\Omega$ is not a slab. The main difficulty is that the family of turned to side nodiods cannot be used anymore. The difficulty was overcome in \cite{Jin3} by constructing a family of auxiliary functions that were used to prove that liftings from subfunctions will be bounded uniformly on any compact subset of $\Omega$. However when $\Omega$ is outside a cone (in the compliment of a cone) or inside a parabola-shaped region, the family of auxiliary functions used in \cite{Jin3} can no longer be used.In this paper, we construct a new family of auxiliary functions so that we can use the Perron's method to prove the existence of solutions to \eqref{eq:problem11}-\eqref{eq:problem12}. As a by product, we can also derive the growth estimates for solutions $u$ to \eqref{eq:problem11}-\eqref{eq:problem12}. For more historical notes and references on prescribed mean curvature equations, we refer readers to \cite{Collin}, \cite{Finn1}, \cite{Finn}, \cite{GT}, \cite{Lopez2}, \cite{Lopez4}, \cite{Wang}. We will consider only those domains that are inside some special regions. The first kind of regions is the compliment of a cone in ${\bf R}^n$ ($n\geq 2$) defined by (we use the notation ${\bf{x}}^{*}=(x_{1}, x_{2}, \cdot \cdot \cdot , x_{n-1})$) $$P(n) = \{ {\bf{x}} \in {\bf R}^n : |x_{n}|< \frac{1}{240n}|{\bf{x}}^{*}| \}.$$ The second kind of regions is a parabola-shaped region defined by $$P(n, \alpha ,b) = \{ {\bf{x}} \in {\bf R}^n : |x_{n}|< b|{\bf{x}}^{*}|^{\alpha } \}.$$ for some fixed positive constants $\alpha$, $b$, $0<\alpha <1$. For a general domain $\Omega$ inside $P(n)$, we can estimate the growth rate of a solution. \begin{theorem} \label{theorem:second} Let $\Omega$ be a domain inside $P(n)$, $|\Lambda ({\bf{x}})|$ satisfy $$|\Lambda ({\bf{x}})| \leq \frac{15(n-1)}{14(n+1)} \frac{1}{|{\bf{x}}^{*}|} \quad\text{on } \Omega , \label{eq:boundoflambda}$$ then any $C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ solution $u$ to \eqref{eq:problem11}-\eqref{eq:problem12} satisfies that on $\Omega$, $$|u({\bf{x}})|\leq \frac{1}{240n}|{\bf{x}}^{*}|+ \sup \{ |\phi ({\bf{p}},q)| : ({\bf{p}},q)\in \partial \Omega, \frac{1}{2}|{\bf{x}}^{*}|\leq |{\bf{p}}|\leq 2 |{\bf{x}}^{*}| \} . \label{eq:boundone}$$ \end{theorem} When $\Omega$ satisfies more geometric conditions, the existence of solutions to \eqref{eq:problem11}-\eqref{eq:problem12} can be proved. First we list a set of conditions that will guarantee a solution to the minimal surface equation with the same boundary data on the same domain: \begin{itemize} \item[(A1)] There is a sequence of subdomains $\Omega_{j}$ such that $\Omega_{j}\subset \Omega_{j+1} \subset \Omega$ for all $j\geq 1$, $\cup \Omega_{j}=\Omega$; \item[(A2)] Each $\Omega_{j}$ is a $C^{2, \gamma }$ bounded domain and has positive mean curvature on $\partial \Omega_{j}$ with respect to the inner normal on $\partial \Omega_{j}$; \item[(A3)] $dist ({\bf{0}}, \Omega \setminus \Omega_{j}) \to \infty$ as $j\to \infty$. \end{itemize} The next condition on $\Omega$ will be used to prove the solution obtained by Perron's method takes boundary data $\phi$ continuously. \noindent{\it{Serrin's condition}}: The mean curvature function $H'$ on $\partial \Omega$ with respect to the inner normal satisfies $$H'> \frac{n}{n-1} |\Lambda ({\bf{x}})| \quad on\quad \partial \Omega . \label{eq:meancurvature}$$ \begin{remark} \label{rmk1.2} \rm Conditions (A1)-(A3) and Serrin's condition (\ref{eq:meancurvature}) are the same as those used in \cite{Jin3}. \end{remark} Here is the first existence result. \begin{theorem} \label{theorem:first} Assume {\rm (A1)--(A3)}, Serrin's condition \eqref{eq:meancurvature} and $\Omega$ is inside $P(n)$. If $\Lambda ({\bf{x}})$ satisfies \eqref{eq:boundoflambda}, then \eqref{eq:problem11}--\eqref{eq:problem12} has a solution $u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$. \end{theorem} When the domains are inside $P(n, \alpha ,b)$, we assume $\Omega$ is not very close to the origin: $$|{\bf{x}}^{*}|\geq (120nb(\frac{3}{2})^{\alpha })^{\frac{1}{1-\alpha}} \quad {\text{for}} \ \ {\text{any}} \quad {\bf{x}} \in \Omega . \label{eq:boundaway0}$$ \begin{remark} \label{rmk1.4} \rm Condition \eqref{eq:boundaway0} is not absolutely necessary, we use it here so that we can state results more clearly. Without \eqref{eq:boundaway0}, the following results are still true as long as $\Lambda ({\bf{x}})$ is bounded appropriately where \eqref{eq:boundaway0} does not hold. \end{remark} The growth estimate now is as follows. \begin{theorem} \label{theorem:fourth} Let $\Omega$ be a domain inside $P(n, \alpha ,b)$. If $\Omega$ satisfies \eqref{eq:boundaway0} and $$|\Lambda ({\bf{x}})| \leq \frac{(n-1)}{56n(n+1)}(\frac{1}{3})^{\alpha } \frac{1}{b|{\bf{x}}^{*}|^{\alpha }} \quad\text{on } \Omega , \label{eq:boundoflambda20}$$ then any $C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ solution $u$ to \eqref{eq:problem11}-\eqref{eq:problem12} satisfies that on $\Omega$ $$|u({\bf{x}})|\leq \frac{1}{2} (\frac{3}{2})^{\alpha} |{\bf{x}}^{*}|^{\alpha} + \sup \{ |\phi ({\bf{p}},q)| : ({\bf{p}},q)\in \partial \Omega, \frac{1}{2}|{\bf{x}}^{*}|\leq|{\bf{p}}|\leq 2 |{\bf{x}}^{*}| \} . \label{eq:boundtwo}$$ \end{theorem} Here is the existence results for domains in $P(n, \alpha, b)$. \begin{theorem} \label{theorem:third} Assume {\rm (A1)--(A3)}, Serrin's condition \eqref{eq:meancurvature} and $\Omega$ is inside $P(n, \alpha ,b)$ satisfying \eqref{eq:boundaway0}. Then if $|\Lambda ({\bf{x}})|$ satisfies \eqref{eq:boundoflambda20}, \eqref{eq:problem11}-\eqref{eq:problem12} has a solution $u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$. \end{theorem} \section{A family of auxiliary functions and growth estimates} In this section, we construct a family of auxiliary functions and derive growth estimates for solutions of \eqref{eq:problem11}-\eqref{eq:problem12}. The construction is adapted from that in \cite{JL2} and \cite{JinKirk} to fit our needs here (in turn, the constructions in \cite{JL2} and \cite{JinKirk} were inspired by \cite{Finn} and \cite{Serrin}). Set $$Q z \equiv \frac{((1+|Dz|^{2}) \delta_{ij} - D_{i}z D_{j}z )}{n+(n-1)|Dz|^{2}}D_{ij} z$$ We first prove the existence of a family auxiliary functions that will suit our needs later. \begin{lemma} \label{lemma:first} For any numbers $M>0$, $H\geq 2$, and any point ${\bf{x}}^{*}_{0}\in R^{n-1}$, there are positive decreasing functions $\chi (t)$ (depending on $n$ only), $h_{a}(t)$ (with the inverse $h_{a}^{-1}$) and a positive increasing function $A(t)$ (depending on $n$, $H$ and $M$ only) such that for any constant $\gamma$, the function $$z=z({\bf{x}})=\gamma +A(H)e^{\chi (H)} -\{ (h_{a}^{-1}(x_{n}+M))^2-|{\bf x}^{*}-{\bf x}^{*}_{0}|^2 \}^{1/2} \label{eq:barrier}$$ satisfies $$Qz \leq -\frac{n-1}{28(n+1) MH} \cdot \frac{(1+|Dz|^{2})^{3/2}}{n+(n-1)|Dz|^{2}} \quad in \quad \Omega_{{\bf x}^{*}_{0},H,M} \label{eq:estimates}$$ where $$\Omega_{{\bf x}^{*}_{0},H,M} =\{{\bf x}:|x_{n}|< M,|{\bf x}^{*}-{\bf x}^{*}_{0}|< h_{a}^{-1}(x_{n}+M) \} . \label{eq:domain}$$ Furthermore $$z({\bf{x}}^{*}_{0},x_{n})\leq \gamma +\frac{M}{H} \quad for \quad -M \leq x_{n} \leq M. \label{eq:boundofz}$$ \end{lemma} \begin{proof} Set $E=\frac{1}{n-1}$, $G=\frac{1}{2n-1}$, $c_{2}=\frac{2+E}{G}=4n+\frac{1}{n-1}$, and $\Phi_{1}(\rho)=\rho^{-2}$ if $0<\rho<1$, $\Phi_{1}(\rho)=c_{2}$ if $\rho \geq 1$. We define a function $\chi$ by $$\chi(\alpha)=\int_{\alpha}^{\infty} \frac{d\rho}{\rho^{3}\Phi_{1}(\rho)} \quad \mbox{for } \alpha>0.$$ It is clear that $\chi (\alpha )$ is a decreasing function with range $(0,\infty).$ Let $\eta$ be the inverse of $\chi.$ Then $\eta$ is a positive, decreasing function with range $(0,\infty)$. For $\alpha >1$, we have $$\chi(\alpha)=\int_{\alpha}^{\infty}\frac{d\rho}{\rho^{3}\Phi_{1}(\rho)} =\int_{\alpha}^{\infty}\frac{d\rho}{c_{2}\rho^{3}} = \frac{1}{2c_{2}} \alpha^{-2} <1. \label{eq:chi2}$$ Thus $$\eta (\beta ) = (2c_{2}\beta)^{-1/2}\quad \text{for } 0<\beta <(2c_{2})^{-1}. \label{eq:beta1}$$ For $H\geq 2$, since $\eta (\chi (H)) =H$ and $\eta$ is decreasing, we have $\eta(\beta)> H$ and $\eta (\beta )= (2c_{2}\beta)^{-1/2}$ for $0<\beta< \chi(H)$. We define a function $A(H)=A(H,M)$ by $$A(H) = 2M (\int_{1}^{e^{\chi(H)}} \eta (\ln t) dt)^{-1} . \label{eq:ah}$$ For the rest of this article, we set $a=A(H)$ and define $$h_{a}(r)=\int_{r}^{ae^{\chi(H)}} \eta (\ln \frac{t}{a} ) dt \quad\text{for } a\le r\le ae^{\chi(H)}.$$ Then $$h_{a}(ae^{\chi (H)})=0, \quad h_{a}(a) =h_{A(H)}(A(H))= 2M. \label{eq:chi3}$$ For $aH, \quad h_{a}''(r)=\frac{1}{r}(\eta(\ln \frac{r}{a} ))^{3}\Phi_{1} (\eta(\ln \frac{r}{a} )). \label{eq:derivativelarge} Thus for$a0$, we define a domain$\Omega_{{\bf x}^{*}_{0},H,M}$in${\bf{x}}$space by (\ref{eq:domain}) and define a function$z=z_{{\bf x}^{*}_{0},H,M}({\bf{x}})$by (\ref{eq:barrier}). It is clear that the function$z$is well defined on$\Omega_{{\bf x}^{*}_{0},H,M}$. Let $$S=((h_{a}^{-1}(x_{n}+M))^2-|{\bf x}^{*}-{\bf x}_{0}^{*}|^2 )^{1/2}. \label{eq:defofs}$$ then for$1\le i\le n-1 $, we have $$\frac{\partial z}{\partial x_i}= \frac{1}{S}(x_i-x_{0i}),\quad \frac{\partial z}{\partial x_{n}}= - \frac{1}{S}h_{a}^{-1} (h_{a}^{-1})'. \label{eq:derivative}$$ Since$h_{a}^{-1}(r)$and$\eta $are decreasing functions, for$H\geq 2$,$|y|\leq M, we have \begin{aligned} 0&< -(h_{a}^{-1})' = \frac{-1}{h_{a}'(h_{a}^{-1}(x_{n}+M))} =\frac{1}{ \eta(\ln(\frac{1}{a}h_{a}^{-1}(x_{n}+M))) } \\ &\leq \frac{1}{\eta (\ln e^{\chi (H)})} =\frac{1}{\eta(\chi(H))}= \frac{1}{H} \quad {\rm{for}} |x_{n}|\le -M. \end{aligned} \label{eq:boundofeta} Then (\ref{eq:boundofz}) follows from the facts thatz({\bf{x}}^{*}_{0},-M)=\gamma +A(H)e^{\chi (H)} -h_{a}^{-1}(0) =\gamma$and $$\frac{\partial z}{\partial x_{n}} ({\bf{x}}^{*}_{0},x_{n})= - \frac{1}{S}h_{a}^{-1} (h_{a}^{-1})'=- (h_{a}^{-1})'\leq \frac{1}{H}.$$ Now if$|{\bf x}^{*}-{\bf x}^{*}_{0}|\geq \frac{1}{2}h_{a}^{-1}(x_{n}+M))$and$H\geq 2$, $$(\frac{\partial z}{\partial x_{n}})^{2} =\frac{1}{S^{2}} (h_{a}^{-1})^{2} ((h_{a}^{-1})')^{2} \leq \frac{1}{S^{2}} (\frac{1}{2}h_{a}^{-1})^{2} \frac{4}{H^{2}} \leq \sum_{i=1}^{n-1} (\frac{\partial z}{\partial x_{i}})^{2} . \label{eq:derivative2}$$ If$|{\bf x}^{*}-{\bf x}^{*}_{0}|\leq \frac{1}{2}h_{a}^{-1}(x_{n}+M))$and$H\geq 2$, then $$S^{2}=(h_{a}^{-1}(x_{n}+M)))^{2} -|{\bf x}^{*}-{\bf x}^{*}_{0}|^{2} \geq \frac{3}{4}(h_{a}^{-1}(x_{n}+M)))^{2},$$ and $$(\frac{\partial z}{\partial x_{n}})^{2} =\frac{1}{S^{2}} (h_{a}^{-1})^{2} ((h_{a}^{-1})')^{2} \leq \frac{1}{S^{2}} (h_{a}^{-1})^{2} \frac{1}{H^{2}} \leq \frac{4}{3H^{2}}\leq 1. \label{eq:derivative3}$$ Therefore, $$(\frac{\partial z}{\partial x_{n}})^{2}\leq \sum_{i=1}^{n-1} (\frac{\partial z}{\partial x_{i}})^{2} +1.$$ We set the notation $$a_{ij} =\frac{(1+|p|^{2})\delta_{ij} -p_{i}p_{j}}{n+(n-1)|p|^{2}}, p_{i}=\frac{\partial z}{\partial x_{i}} \quad 1\leq i, j \leq n.$$ Then$|p_{n}|^{2}\leq \sum_{i=1}^{n-1}p_{i}^{2} +1$and $$a_{nn}=\frac{1+\sum_{i=1}^{n-1}p_{i}^{2}}{n+(n-1)|p|^{2}} \geq \frac{1+\sum_{i=1}^{i=n-1}p_{i}^{2}}{2n-1+2(n-1)\sum_{i=1}^{i=n-1}p_{i}^{2}} \geq \frac{1}{2n-1}=G \label{eq:derivative4}$$ and $$\sum_{i,j=1}^{n} a_{ij} \frac{\partial z}{\partial x_{i}} \frac{\partial z}{\partial x_{j}} =\frac{|p|^{2}}{n+(n-1)|p|^{2}} \leq \frac{1}{n-1} =E. \label{eq:derivative5}$$ Thus on$\Omega_{{\bf{x}}^{*}_{0}, H,M}, we have \begin{align*} Qz&=\sum_{i,j=1}^{n} a_{ij}D_{ij}z \\ &= \frac{1}{S}\sum_{i=1}^{n-1} a_{ii} +\frac{1}{S^{3}}\sum_{i,j=1}^{n-1} a_{ij}(x_{i}-x_{i}^{0})(x_{j}-x_{j}^{0}) -\frac{1}{S^{3}}\sum_{i=1}^{n-1} a_{in}(x_{i}-x_{i}^{0})h_{a}^{-1} (h_{a}^{-1})' \\ &\quad - \frac{1}{S}a_{nn}((h_{a}^{-1})^{2} + h_{a}^{-1} (h_{a}^{-1})'') + \frac{1}{S^{3}}a_{nn}(h_{a}^{-1})^{2} ((h_{a}^{-1})')^{2}\\ &= \frac{1}{S} \big\{ 1-a_{nn}+\sum_{i,j=1}^{n} a_{ij} \frac{\partial z}{\partial x_{i}} \frac{\partial z}{\partial x_{j}} -a_{nn} ((h_{a}^{-1})^{2} + h_{a}^{-1} (h_{a}^{-1})'') \big\} \\ & \leq \frac{1}{S}\big\{ 1+\sum_{i,j=1}^{n} a_{ij} \frac{\partial z}{\partial x_{i}} \frac{\partial z}{\partial x_{j}} -a_{nn} h_{a}^{-1} (h_{a}^{-1})'' \big\} \\ &\leq \frac{1}{S}\{ 1+E - Gh_{a}^{-1} (h_{a}^{-1})'' \}= \frac{-1}{S} \end{align*} (by (\ref{eq:equationofinverse})), (\ref{eq:derivative4}), (\ref{eq:derivative5}) and the definition of\Phi $). Then (\ref{eq:estimates}) follows from the following inequality $$\frac{n-1}{28(n+1)MH} \cdot \frac{(1+|Dz|^{2})^{3/2}}{n+(n-1)|Dz|^{2}} \leq \frac{1}{S}. \label{eq:lastone}$$ To prove (\ref{eq:lastone}), since$\chi (H)<1$for$H\geq 2, we have \begin{align*} &\frac{(1+|Dz|^{2})^{3/2}}{n+(n-1)|Dz|^{2}}\\ &\leq \frac{1}{n-1}(1+|Dz|^{2})^{1/2} \\ &=\frac{1}{n-1} (1+\frac{1}{S^{2}} (|{\bf{x}}^{*} -{\bf{x}}_{0}^{*}|^{2} + (h_{a}^{-1})^{2} ((h_{a}^{-1})')^{2}))^{1/2} \quad \text{by \eqref{eq:derivative}} \\ &= \frac{1}{(n-1)S} (S^{2}+|{\bf{x}}^{*} -{\bf{x}}^{*}_{0}|^{2} + (h_{a}^{-1})^{2} ((h_{a}^{-1})')^{2})^{1/2} \\ &=\frac{1}{(n-1)S} ((h_{a}^{-1})^{2} + (h_{a}^{-1})^{2} ((h_{a}^{-1})')^{2})^{1/2} \quad \text{by \eqref{eq:defofs}}\\ &=\frac{1}{(n-1)S} (h_{a}^{-1})(1+((h_{a}^{-1})')^{2})^{1/2} \\ &\leq \frac{1}{(n-1)S} (h_{a}^{-1})(1+\frac{1}{H^{2}})^{1/2} \leq \frac{1}{(n-1)S} A(H)e^{\chi (H)}(1+\frac{1}{4})^{1/2} \quad \text{by \eqref{eq:valueofinverseh}, \eqref{eq:boundofeta}}\\ &\leq \frac{1}{(n-1)S} (\frac{5}{4})^{1/2} 2c_{2}e^{\chi (H)}MH \leq \frac{1}{(n-1)S} c_{2}5^{1/2} e MH \quad \text{by \eqref{eq:ah2}}\\ &=\frac{1}{(n-1)S} (4n+\frac{1}{n-1}) 5^{1/2} e MH \leq \frac{28(n+1)}{n-1} MH \frac{1}{S} \quad \text{by the definition ofc_{2}}. \end{align*} \end{proof} \begin{lemma} \label{lemma:second} Let\phi $be a continuous function defined on$\partial \Omega$. For any${\bf{x}}^{*}_{0} \in R^{n-1}$, we set $$\gamma = \gamma({\bf{x}}^{*}_{0}) =\sup \{ |\phi ({\bf{x}})| : {\bf{x}}\in \partial \Omega, \quad \frac{1}{2}|{\bf{x}}^{*}_{0}|\leq |{\bf{x}}^{*}|\leq \frac{3}{2}|{\bf{x}}^{*}_{0}| \} . \label{eq:definitionofgamma}$$ For any${\bf{x}}^{*}_{0} \in R^{n-1}$such that$({\bf{x}}^{*}_{0},x_{n})\in \Omega$for some$x_{n}$, in the function$z=z_{{\bf{x}}^{*}_{0}} $defined in (\ref{eq:barrier}), we set $$\gamma=\gamma ({\bf{x}}^{*}_{0}), \quad H=2, \quad M=\frac{1}{120n}|{\bf{x}}^{*}_{0}|. \label{eq:setofmh}$$ Then$z=z_{{\bf{x}}^{*}_{0}} $satisfies $$Qz\leq -n \Lambda_{0} ({\bf{x}}) \frac{(1+|Dz|^{2})^{3/2}}{n+(n-1)|Dz|^{2}} \quad in \quad \Omega_{{\bf x}_{0},H,M} \cap P(n) \label{eq:supersolution}$$ where $$\Lambda_{0} ({\bf{x}}) = \frac{15(n-1)}{14(n+1)|{\bf{x}}^{*}|} . \label{eq:boundoflambda0}$$ Furthermore \begin{itemize} \item[(i)]$z({\bf{x}}^{*}_{0},x_{n})\leq \frac{1}{240n} |{\bf{x}}^{*}_{0}| + \gamma({\bf{x}}^{*}_{0})$for$|x_{n}|0$be a continuous function on$\overline{\Omega}$, for each open set$O\in \Pi$, we define a new function$M_{O}(v)$, called the lifting of$v$over$O$as follows: $$M_{O}(v)({\bf{x}})=v({\bf{x}}) \quad\text{if } {\bf{x}}\in \Omega\setminus O, \quad M_{O}(v)({\bf{x}},y)=w({\bf{x}}) \quad\text{if } {\bf{x}}\in O$$ where$w({\bf{x}})$is the solution of the boundary-value problem \begin{gather} ((1+|Dw|^{2}) \delta_{ij} - D_{i}w D_{j}w )D_{ij} w = n \Lambda ({\bf{x}}) (1+|Dw|^{2})^{3/2} \quad\text{in } O, \label{eq:lift1} \\ w=v \quad\text{on } \partial O \,. \label{eq:lift2} \end{gather} \begin{remark} \rm By (\ref{eq:meancurvatureofo}), Lemma \ref{lemma:growth1} and \cite{Serrin} or \cite[Theorem 16.9]{GT}, there is a unique solution$w\in C^{2}(O)\cap C^{0}(\overline{\Omega})$to (\ref{eq:lift1})-(\ref{eq:lift2}). Thus$M_{O}(v)$is well defined. \end{remark} We define a class$\Xi $of functions$v$, called subfunctions, such that: \begin{enumerate} \item$v\in C^{0}(\overline{\Omega })$and$v\leq \phi $on$\partial \Omega$; \item For any$O \in \Pi $,$v\leq M_{O}(v)$; \item$v\leq z_{{\bf{x}}^{*}_{0}} $on$\Omega \cap \Omega_{{\bf{x}}^{*}_{0}, M, H} $for any${\bf{x}}^{*}_{0} \in R^{n-1}$such that$({\bf{x}}^{*}_{0},x_{n})\in \Omega$for some$x_{n}$, where$z_{{\bf{x}}^{*}_{0}}$are those functions defined in Lemma \ref{lemma:second}. \end{enumerate} Now we prove some properties for subfunctions in the class$\Xi$. \begin{lemma}\label{lemma:fourth} If$v_{1}\leq v_{2}$, then$M_{O}(v_{1})\leq M_{O}(v_{2})$for any$O\in \Pi $. \end{lemma} \begin{proof} Let$w_{1}$,$w_{2}$be the solutions of the following two problems, respectively: \begin{gather*} ((1+|Dw_{k}|^{2}) \delta_{ij} - D_{i}w_{k} D_{j}w_{k} ) D_{ij} w_{k} = n \Lambda ({\bf{x}}) (1+|Dw_{k}|^{2})^{3/2} {\rm{in}} O, \\ w_{k}=v_{k} \quad \text{on } \partial O, \quad k=1,2. \end{gather*} Since$w_{1}=v_{1}\leq v_{2}=w_{2}$on$\partial O$, by a comparison principle for quasilinear elliptic equations (e.g. see \cite[Theorem 10.1]{GT}), we have$w_{1}\leq w_{2}$on$O$. On$\Omega \setminus O$,$M_{O}(v_{1})=v_{1}$,$M_{O}(v_{2})=v_{2}$. Thus$M_{O}(v_{1})\leq M_{O}(v_{2})$. \end{proof} \begin{lemma} \label{lemma:fifth} If$v_{1}\in \Xi$,$v_{2}\in \Xi$, then$\max \{ v_{1}, v_{2} \} \in \Xi$. \end{lemma} \begin{proof} If$v_{1}\in \Xi$,$v_{2}\in \Xi$, then$\max \{ v_{1}, v_{2} \}\in C^{0}(\overline{\Omega })$, and$\max \{ v_{1}, v_{2} \}\leq \phi $on$\partial \Omega$. It is also clear that$\max \{ v_{1}, v_{2} \}\leq z_{{\bf{x}}^{*}_{0}}$on$\Omega_{{\bf{x}}^{*}_{0}, M, H } \cap \Omega$. Since$v_{1}\leq \max \{ v_{1}, v_{2} \}$,$v_{2} \leq \max \{ v_{1}, v_{2}\}$, we have (by Lemma \ref{lemma:fourth}) that for any$O\in \Pi $, $$M_{O}(v_{1}) \leq M_{O}(\max \{ v_{1}, v_{2}\} ),\quad M_{O}(v_{2}) \leq M_{O}(\max \{ v_{1}, v_{2}\} ).$$ Since$v_{1}\in \Xi $and$v_{2}\in \Xi $imply$v_{1} \leq M_{O}(v_{1})$,$v_{2} \leq M_{O}(v_{2})$, we have$\max \{ v_{1}, v_{2} \}\leq M_{O}(\max \{ v_{1}, v_{2}\})$. Thus$\max \{ v_{1}, v_{2} \}\in \Xi$. \end{proof} \begin{lemma} \label{lemma:sixth} If$v\in \Xi $, then$M_{O}(v)\in \Xi$for any$O\in \Pi$. \end{lemma} \begin{proof} By the definition of$M_{O}( v)$, it is clear that$M_{O}( v)\in C^{0}(\overline{\Omega} )$and$M_{O}(v) \leq \phi $on$\partial \Omega $. First we show that for any$O_{1}\in \Pi$, $$M_{O}( v)({\bf{x}})\leq M_{O_{1}}(M_{O}(v))({\bf{x}}). \label{eq:mv}$$ We need to prove only that (\ref{eq:mv}) is true for${\bf{x}}\in O_{1}$. Since$v\leq M_{O}( v)$on$\Omega$, we have (by Lemma \ref{lemma:fourth})$M_{O_{1}}( v)\leq M_{O_{1}}(M_{O}( v))$. Combining this with$v\leq M_{O_{1}}( v)$, we have$v\leq M_{O_{1}}(M_{O}(v))$. Thus for${\bf{x}}\in O_{1} \setminus O$, $$M_{O}( v)({\bf{x}})=v({\bf{x}}) \leq M_{O_{1}}(M_{O}(v))({\bf{x}}). \label{eq:boundary1}$$ That is, (\ref{eq:mv}) is true on$O_{1} \setminus O$, Now for$\Omega_{1}=O_{1} \cap O$, if we set $$M_{O}( v)=w_{1}, \quad M_{O_{1}}(M_{O}(v))=w_{2},$$ we have that on$\Omega_{1}$,$k=1,2$, $$((1+|Dw_{k}|^{2}) \delta_{ij} - D_{i}w_{k} D_{j}w_{k} )D_{ij} w_{k} = n \Lambda ({\bf{x}}) (1+|Dw_{k}|^{2})^{3/2} .$$ On$\partial \Omega_{1}$,$w_{1}\leq w_{2}$on$O_{1}\cap \partial O$by (\ref{eq:boundary1}) and$w_{1}\leq w_{2}$on$\partial O_{1}\cap O$since (\ref{eq:mv}) is true on$\Omega\setminus O_{1}$. Then a comparison argument implies$w_{1}\leq w_{2}$on$\Omega_{1}$. Thus (\ref{eq:mv}) is true on$O_{1} \cap O$and on$O_{1}$. \end{proof} Now we prove that$M_{O}(v)\leq z_{{\bf{x}}^{*}_{0}}$on$\Omega_{{\bf{x}}^{*}_{0},M,H}\cap \Omega $. Since$v\in \Xi$,$v\leq z_{{\bf{x}}^{*}_{0}}$on$\Omega_{{\bf{x}}^{*}_{0},M,H}\cap \Omega $. Thus by the definition of$M_{O}(v)$, we only need to show$M_{O}(v)\leq z_{{\bf{x}}^{*}_{0}}$on$\Omega_{{\bf{x}}^{*}_{0},M,H}\cap O$. If$O$does not intersect with$\Omega_{{\bf{x}}^{*}_{0},M,H}$, the conclusion is trivial. In the case that$O$is at least partly covered by$\Omega_{{\bf{x}}^{*}_{0},M,H}$.$M_{O}(v)- z_{{\bf{x}}^{*}_{0}}$cannot achieve its maximum value in$\overline{\Omega_{{\bf{x}}^{*}_{0},M,H}\cap O}$on$\partial \Omega_{{\bf{x}}^{*}_{0},M,H} \cap O$since the directional derivative of$z_{{\bf{x}}^{*}_{0}}$with respect to outer normal is$+\infty$on$\partial \Omega_{{\bf{x}}^{*}_{0},M,H}\cap O $by (iii) in Lemma \ref{lemma:second}. Furthermore since$z_{{\bf{x}}^{*}_{0}}$satisfies (\ref{eq:supersolution}) and$|\Lambda ({\bf{x}})|\leq \Lambda_{0}({\bf{x}})$by \eqref{eq:boundoflambda} and (\ref{eq:boundoflambda0}), a comparison argument concludes that$M_{O}(v)- z_{{\bf{x}}^{*}_{0}}$cannot achieve a local maximum inside$\Omega_{{\bf{x}}^{*}_{0},M,H}\cap O$. Thus$M_{O}(v) -z_{{\bf{x}}^{*}_{0}}$achieves its maximum value in$\overline{\Omega_{{\bf{x}}^{*}_{0},M,H}\cap O}$on$\Omega_{{\bf{x}}^{*}_{0},M,H}\cap \partial O$. Then$M_{O}(v) -z_{{\bf{x}}^{*}_{0}}\leq 0$on$\Omega_{{\bf{x}}^{*}_{0},M,H}\cap O$follows from$M_{O}(v) -z_{{\bf{x}}^{*}_{0}}=v-z_{{\bf{x}}^{*}_{0}}\leq 0$on$\Omega_{{\bf{x}}^{*}_{0},M,H}\cap \partial O$. %\end{proof} Now we will show that$\Xi$is not empty by proving the existence of a solution to the minimal surface equation with the same boundary-value and on the same domain. \begin{lemma} \label{lemma:seventh} If$v\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$is a solution of the problem $$((1+|Dv|^{2}) \delta_{ij} - D_{i}v D_{j}v )D_{ij} v = 0 {\rm{in}} \Omega , v=\phi on \partial \Omega . \label{eq:subsolution221}$$ Then for any$({\bf{x}}^{*}_{0},x_{n})\in \Omega$, $$|v| \leq z_{{\bf{x}}^{*}_{0}}\quad\text{on } \Omega_{{\bf{x}}^{*}_{0},M,H} \cap \Omega \label{eq:boundofminimal}$$ \end{lemma} The proof of the above lemma is just a special case of Lemma \ref{lemma:growth1} with$\Lambda ({\bf{x}})=0$. \begin{lemma} \label{lemma:eigth} Assume {\rm (A1)--(A3)}. Then the boundary-value problem $$((1+|Dv|^{2}) \delta_{ij} - D_{i}v D_{j}v )D_{ij} v = 0 \quad\text{in } \Omega ,\quad v=\phi \quad\text{on } \partial \Omega . \label{eq:subsolution1}$$ has a solution$u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$. \end{lemma} \begin{proof} This is \cite[Lemma 4.5 ]{Jin3} (Though a slight difference should be noted there. That is, the bound for solutions of the minimal surface equation is given by (\ref{eq:boundofminimal}) which will play the same role as the Lemma 4.4 in \cite{Jin3} (By the way, the Lemma 3.1 quoted in the proof of Lemma 4.5 in \cite{Jin3} should be Lemma 4.4 in \cite{Jin3}). \end{proof} Now we prove the Theorem \ref{theorem:first}. We set $$u({\bf{x}})=\sup \{ v({\bf{x}}) : v\in \Xi \}, {\bf{x}}\in \overline{\Omega}.$$ We first consider the case that$\Lambda ({\bf{x}})\leq 0$on$\Omega $. For such a choice of$\Lambda ({\bf{x}})$, we will show that$u$is in$C^{0}(\overline{\Omega})\cap C^{2}(\Omega)$satisfying \eqref{eq:problem11}-\eqref{eq:problem12}. It is well known and standard (for example, see \cite{CH}) that by Perron's method, we can prove that$u$is in$C^{2}(\Omega )$and satisfies \eqref{eq:problem11}. Indeed, let${\bf{x}}_{1}\in \Omega $. By the definition of$u({\bf{x}}_{1})$, there is a sequence of functions$v_{i}$in$\Xi $such that $$u({\bf{x}}_{1})=\lim_{i\to \infty } v_{i}({\bf{x}}_{1}).$$ Let$v_{0} $be a solution of (\ref{eq:subsolution1}). Since$\Lambda ({\bf{x}})\leq 0$on$\Omega $, by Lemma \ref{lemma:seventh}, it is easy to check that$v_{0}\in \Xi$. By Lemma \ref{lemma:fifth} and replacing$v_{i}$by$\max \{ v_{i}, v_{0} \}$, we may assume that$v_{i}\geq v_{0} $on$\Omega$. Let$O$be an open set in$\Pi$such that${\bf{x}}_{1}\in O$. We replace$v_{i}$by$M_{O}(v_{i})$. Then we have a sequence of functions$z_{i}$defined on$O$satisfying \begin{gather*} u({\bf{x}}_{1})=\lim_{i\to \infty } z_{i}({\bf{x}}_{1}) , \\ ((1+|Dz_{i}|^{2}) \delta_{pq} - D_{p}z_{i} D_{q}z_{i}) D_{pq} z_{i} = n \Lambda ({\bf{x}}) (1+|Dz_{i}|^{2})^{3/2} \quad\text{on } O, \\ z_{i}=v_{i} \quad\text{on } \partial O. \end{gather*} Since for all$i$, if$O\cap \Omega_{{\bf{x}}^{*}_{0},M,H}$is not empty, $$v_{0} \leq v_{i}\leq z_{i} \leq z_{{\bf{x}}^{*}_{0}} \quad\text{on } O\cap \Omega_{{\bf{x}}^{*}_{0},M,H},$$ and we can cover$O$by finitely many such domains$\Omega_{{\bf{x}}^{*}_{0},M,H}$, thus there is a number$K_{3}$independent of$i$, such that for all$i$, $$v_{0} \leq z_{i}\leq K_{3} \quad\text{in } O.$$ By \cite[Corollarys 16.6, 16.7]{GT}, there is a subsequence of$z_{i}$, for convenience still denoted by$z_{i}$, converges to a$C^{2}(O)$function$z(x)$in$C^{2}(O)$. Thus$z(x)$satisfies $$((1+|Dz|^{2}) \delta_{pq} - D_{p}z D_{q}z )D_{pq} z= n \Lambda ({\bf{x}}) (1+|Dz|^{2})^{3/2} \quad\text{on } O.$$ Note that$u({\bf{x}}_{1})=z({\bf{x}}_{1})$and$u({\bf{x}})\geq z({\bf{x}})$on$O$. We claim that$u=z$on$O$. Indeed, if there is another point${\bf{x}}_{2} \in O$such that$u({\bf{x}}_{2})$is not equal to$z({\bf{x}}_{2})$, we must have$u({\bf{x}}_{2})>z({\bf{x}}_{2})$. Then there is a function$u_{0}\in \Xi $, such that $$z({\bf{x}}_{2})< u_{0}({\bf{x}}_{2})\leq u({\bf{x}}_{2}).$$ Now the sequence$\max \{ u_{0}, M_{O}(v_{i}) \}$satisfying $$v_{i} \leq \max \{ u_{0}, M_{O}(v_{i}) \} \leq u .$$ Then similar to the way we obtain$z$,$M_{O}(\max \{ u_{0}, M_{O}(v_{i}) \})$will produce a$C^{2}$function$z_{1}$satisfying \begin{gather*} ((1+|Dz_{1}|^{2}) \delta_{pq} - D_{p}z_{1} D_{q}z_{1} ) D_{pq} z_{1} = n \Lambda ({\bf{x}}) (1+|Dz_{1}|^{2})^{3/2} \quad\text{on } O,\\ z\leq z_{1} \quad\text{on } O, \quad z({\bf{x}}_{2}) \frac{n}{n-1} |\Lambda ({\bf{x}})| \quad\text{on } \partial \Omega_{1}. \label{eq:boundaryofsmall} Since$\Omega_{1}$can be covered by finitely many$\Omega_{{\bf{x}}^{*}_{0}, M, H}$, there is a number$K_{4}>0$, such that for all$v\in \Xi$, $$v\leq K_{4} \quad\text{on } {\overline{\Omega}}_{1} . \label{eq:boundofsmall}$$ Now on$\partial \Omega_{1}$, we choose a smooth function$\phi^{*}$as follows.$\phi^{*}=K_{4}$on$\partial \Omega_{1}\cap \Omega$.$\phi^{*}=\phi $in a neighborhood of${\bf{x}}_{1}$in$\partial \Omega_{1}$and$\phi^{*}\geq \phi $on the rest of$\partial \Omega_{1}$(since (\ref{eq:boundofsmall}) implies$\phi \leq K_{4}$on$\partial \Omega_{1}\cap \partial \Omega$, this is possible). Now we consider the boundary-value problem $$((1+|Du|^{2}) \delta_{ij} - D_{i}u D_{j}u ) D_{ij} u = n \Lambda ({\bf{x}}) (1+|Du|^{2})^{3/2} \quad\text{on } \Omega_{1}, \label{eq:small}$$ $$u=\phi^{*} \quad\text{on } \partial \Omega_{1} . \label{eq:small22}$$ From (\ref{eq:boundaryofsmall}), Lemma \ref{lemma:growth1} and \cite{Serrin} or \cite[Theorem 16.9]{GT}, (\ref{eq:small})-(\ref{eq:small22}) has a solution$u_{1}\in C^{2}(\Omega_{1})\cap C^{0}({\overline{\Omega}}_{1})$. From the definition of$u_{1}$, (\ref{eq:boundofsmall}) and the fact that$v=\phi $on$\partial \Omega$for any$v\in \Xi$, a comparison argument shows that for any$v\in \Xi$, $$M_{\Omega_{1}}(v) \leq u_{1} \quad\text{on } \Omega_{1}\quad {\text{for any}} \quad v\in \Xi.$$ Therefore, $$u\leq u_{1} \quad\text{on } \Omega_{1}. \label{eq:upperhalf}$$ Since we always have $$u\geq v_{0} \quad\text{on } \Omega$$ for the solution$v_{0}$of (\ref{eq:subsolution1}), we have $$v_{0}\leq u\leq u_{1} \quad\text{on } \Omega_{1} . \label{eq:whole}$$ Then the continuity of$u$at${\bf{x}}_{1}$follows from the fact that$v_{0}=u_{1}=\phi $on a neighborhood of${\bf{x}}_{1}$in$\partial \Omega$and both$v_{0}$and$u_{1}$are continuous in a neighborhood of${\bf{x}}_{1}$in$\overline{\Omega}$. Since${\bf{x}}_{1}\in \partial \Omega$can be arbitrary, we have$u\in C^{0}(\overline{\Omega})$. Thus under the additional assumption that$\Lambda ({\bf{x}})\leq 0$on$\Omega$, we have proved Theorem \ref{theorem:first}. In the case that$\Lambda ({\bf{x}})\geq 0$on$\Omega$, repeating above proof, we can find a function$u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$satisfying \begin{gather*} ((1+|Du|^{2}) \delta_{ij} - D_{i}u D_{j}u ) D_{ij} u = -n\Lambda ({\bf{x}}) (1+|Du|^{2})^{3/2} \quad\text{on } \Omega , \\ u=-\phi \quad\text{on } \partial \Omega . \end{gather*} Then$-u$will satisfy \eqref{eq:problem11}-\eqref{eq:problem12}. In the general case of$\Lambda ({\bf{x}})$, we first find a function$u_{0}\in C^{1}(\Omega)\cap C^{0}(\overline{\Omega})$satisfying \begin{gather*} ((1+|Du|^{2}) \delta_{ij} - D_{i}u D_{j}u ) D_{ij} u = n|\Lambda ({\bf{x}})| (1+|Du|^{2})^{3/2} \quad\text{on } \Omega , \\ u=\phi \quad\text{on } \partial \Omega . \end{gather*} In the proof for the case that$\Lambda\leq 0$, we replace$v_{0}$(the solution of (\ref{eq:subsolution1})) by$u_{0}$, without changing the rest of the proof, now we will obtain a function$u\in C^{1}(\Omega)\cap C^{0}(\overline{\Omega})$satisfies \eqref{eq:problem11}-\eqref{eq:problem12}. This completes the proof for Theorems \ref{theorem:first}. \subsection*{Acknowledgements} The author would like to thank Professor J. Serrin for his valuable suggestions. \begin{thebibliography}{00} \bibitem{Bernstein} S. Berstein; Sur les surfaces definies au moyen de leur courbure moyenne et totale, {\em Ann. Scuola Norm. Sup. Pisa}, 27 (1910), 233-256. \bibitem{Collin} P. 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