# Smooth Solutions to Asymptotic Plateau Type Problem in Hyperbolic Space

Zhenan Sui  Institute for Advanced Study in Mathematics of HIT, Harbin
Institute of Technology, Harbin, China<EMAIL_ADDRESS>and  Wei Sun  Institute
of Mathematical Sciences, ShanghaiTech University, Shanghai, China
<EMAIL_ADDRESS>
###### Abstract.

We investigate on the existence of smooth complete hypersurface with
prescribed Weingarten curvature and asymptotic boundary at infinity in
hyperbolic space under the assumption that there exists an asymptotic
subsolution. We give an affirmative answer for the case $k=n$ when the
asymptotic boundary $\Gamma$ bounds a uniformly convex domain, and for $k<n$
when $\Gamma$ bounds a disk, utilizing Pogorelov type interior second order
estimate. Our result complements our previous work [13, 14], and generalizes
the asymptotic Plateau type problem to non-constant prescribed curvature case.

###### 2010 Mathematics Subject Classification:

Primary 53C21; Secondary 35J65, 58J32

##  1\. Introduction

In this paper, we shall continue our study of finding smooth hypersurfaces to
asymptotic Plateau type problem in hyperbolic space, which extends our
previous work [13, 14]. As before, we take the half space model for hyperbolic
space

$\mathbb{H}^{n+1}=\\{(x,x_{n+1})\in\mathbb{R}^{n+1}\big{|}x_{n+1}>0\\},$

endowed with the metric

$ds^{2}=\frac{1}{x_{n+1}^{2}}\sum_{i=1}^{n+1}dx_{i}^{2}.$

Let $\psi$ be a given smooth positive function defined in $\mathbb{H}^{n+1}$,
and $\Gamma=\\{\Gamma_{1},\ldots,\Gamma_{m}\\}$ be a disjoint collection of
smooth closed $(n-1)$ dimensional submanifolds at

$\partial_{\infty}\mathbb{H}^{n+1}:=\mathbb{R}^{n}\times\\{0\\}\cong\mathbb{R}^{n}.$

We want to seek a smooth complete connected admissible vertical graph
$\Sigma=\\{(x,u(x))|x\in\Omega\\}$ satisfying

(1.1) $\left\\{\begin{aligned}
f(\kappa[u])=&\,\sigma_{k}^{\frac{1}{k}}(\kappa)=\psi(x,u)\quad&\mbox{in}\,\,\Omega,\\\
u=&\,0\quad&\mbox{on}\,\,\Gamma,\end{aligned}\right.$

where $\kappa=(\kappa_{1},\ldots,\kappa_{n})$ are the hyperbolic principal
curvatures of $\Sigma$ with respect to the upward normal,

$\sigma_{k}(\kappa)=\sum_{1\leq i_{1}<\ldots<i_{k}\leq
n}\kappa_{i_{1}}\cdots\kappa_{i_{k}}$

is the $k$th-Weingarten curvature defined on $k$-th Gårding’s cone

$\Gamma_{k}=\\{\kappa\in\mathbb{R}^{n}|\sigma_{j}(\kappa)>0,\,j=1,\ldots,k\\},$

and $\Omega$ is the bounded domain enclosed by $\Gamma$ on $\mathbb{R}^{n}$.
$\Sigma$ is said to be admissible if $\kappa\in\Gamma_{k}$. When $k=1,2,n$,
$\sigma_{k}(\kappa)$ is the so-called mean curvature, scalar curvature and
Gauss curvature. In the special case $k=n$, we shall use “strictly locally
convex” instead of “admissible” when $\kappa\in\Gamma_{n}$. Besides, we say
that $\Sigma$ is locally convex if $\kappa\in\overline{\Gamma_{n}}$.

Asymptotic Plateau type problem (1.1) is more difficult than common Dirichlet
problems due to the singularity at $\Gamma$. Because of this, it resembles
those problems on noncompact domains. When $\psi=\sigma\in(0,1)$ is a
prescribed constant, Guan, Spruck, Szapiel and Xiao [5, 8, 6, 7, 9] developed
some effective technique for problem (1.1). They analyze the approximating
Dirichlet problem

(1.2) $\left\\{\begin{aligned}
f(\kappa[u])=&\,\sigma\quad&\mbox{in}\,\,\Omega,\\\
u=&\,\epsilon\quad&\mbox{on}\,\,\Gamma,\end{aligned}\right.$

where $f$ is in general setting, satisfies certain assumptions, and is
normalized, and $\epsilon$ is a small positive constant. Because
$\psi=\sigma\in(0,1)$ is a constant, there is a natural subsolution
$\underline{u}=\epsilon$ during the continuity process related to (1.2), and
the estimates to (1.2) can be independent of $\epsilon$.

When $\psi$ is not a constant, we no longer have a natural subsolution during
the continuity process, and most of the methods in [5, 8, 6, 7, 9] do not work
in this generality. In [13, 14], we started a new method: assume that there
exists an asymptotic subsolution to (1.1), namely, there exists an admissible
$\underline{u}\in C^{4}(\Omega)\cap C^{0}(\overline{\Omega})$ such that

(1.3) $\left\\{\begin{aligned}
f(\kappa[\underline{u}])\geq&\,\psi(x,\underline{u})\quad&\mbox{in}\,\,\Omega,\\\
\underline{u}=&\,0\quad&\mbox{on}\,\,\Gamma.\end{aligned}\right.$

Then we make use of the level set of $\underline{u}$ to construct
approximating Dirichlet problem:

(1.4) $\left\\{\begin{aligned}
f(\kappa[u])=&\,\psi(x,u)\quad&\mbox{in}\,\,\Omega_{\epsilon},\\\
u=&\,\epsilon\quad&\mbox{on}\,\,\Gamma_{\epsilon},\end{aligned}\right.$

where

$\Gamma_{\epsilon}=\\{x\in\Omega\,\big{|}\,\underline{u}(x)=\epsilon\\},\quad\Omega_{\epsilon}=\\{x\in\Omega\,\big{|}\,\underline{u}(x)>\epsilon\\}.$

We also assume that $\Gamma_{\epsilon}$ is a regular boundary of
$\Omega_{\epsilon}$ when $\epsilon>0$ is sufficiently small, namely,
$\Gamma_{\epsilon}$ has dimension $n-1$, $\Gamma_{\epsilon}\in C^{4}$ and
$|D\underline{u}|>0$ on $\Gamma_{\epsilon}$. Our idea is like this. The first
step is to prove the existence of solutions to (1.4), which needs the
establishment of $C^{2}$ estimates to (1.4). The feature is that we allow the
dependence on $\epsilon$. Consequently, techniques for common Dirichlet
problems can be applied. The next step is to derive certain uniform interior
estimates up to second order which are independent of $\epsilon$ for solutions
to (1.4). Combined with Evans-Krylov interior estimates and diagonal process,
we can then obtain smooth solutions to (1.1).

We have finished the first step in [13, 14] under certain assumptions. In
addition, we obtained uniform interior estimates up to first order for
solution $u^{\epsilon}$ to (1.4) in these papers, that is, for any
sufficiently small fixed $\epsilon_{0}>0$, we have

(1.5) $\|u^{\epsilon}\|_{C^{1}(\overline{\Omega_{\epsilon_{0}}})}\leq
C,\quad\forall\,\,0<\epsilon<\frac{\epsilon_{0}}{2}.$

Here and throughout this paper, $C$ denotes a positive constant which is
independent of $\epsilon$, but may depend on $\epsilon_{0}$.

The difficulty lies in the establishment of uniform interior estimates of
second order for $u^{\epsilon}$. As known from literature, the endeavor to
derive interior $C^{2}$ estimates fails in general context. For example, from
the counterexamples of Pogorelov [11] and Urbas [16], we know that the pure
interior $C^{2}$ estimates do not hold when $k\geq 3$. Because of this fact,
we investigated prescribed scalar curvature equations in [13], using the idea
of Guan-Qiu [10], who established interior curvature estimate for strictly
locally convex solutions to prescribed scalar curvature equations in Euclidean
space. Subsequently, we studied the next best thing in [14]: to obtain a
general existence result for locally Lipschitz continuous weak admissible
solutions to (1.1).

Despite the lack of pure interior $C^{2}$ estimate for general prescribed
curvature equations, we seek the possibility of other types of interior
$C^{2}$ estimates to accomplish our second step. Recently, we observe that
Pogorelov type interior $C^{2}$ estimate will suffice to give our desired
uniform interior second order estimate with the help of some suitable nested
domains (see (2.11) and (3.6)), which are built by means of solution to the
associated homogeneous equation

(1.6) $\left\\{\begin{aligned} f(\kappa[v])=&\,0\quad&\mbox{in}\,\,\Omega,\\\
v=&\,0\quad&\mbox{on}\,\,\Gamma,\end{aligned}\right.$

as well as the asymptotic subsolution $\underline{u}$. This observation is
motivated by the work of Trudinger-Urbas [15], who studied Pogorelov type
interior second order estimate for Monge-Ampère equations with general
Dirichlet boundary values over uniformly convex domains. However, their
estimate can not be directly applied to solution $u^{\epsilon}$ of (1.4) with
$k=n$, because it relies on the convexity of the domain and the $C^{2}$ norm
of the boundary data (in hyperbolic space, it would also depend on the lower
bound of $u$), and consequently may not bring in a uniform interior bound. Our
strategy is: rather than derive Pogorelov type interior second order estimate
for $u^{\epsilon}$ on $\Omega_{\epsilon}$, we consider this estimate on
$\Omega^{\epsilon}_{\epsilon_{0}}$ (see section 2 and 3 for the definitions),
which are uniformly away from $\Gamma$. As a result, we obtain the uniform
interior second order estimate (2.19) when $k=n$, which in turn gives the
first main result of this paper on prescribed Gauss curvature equations.

######  Theorem 1.1.

For $k=n$ and $0<\psi(x,u)\in C^{\infty}(\mathbb{H}^{n+1})$, suppose that
there exists a strictly locally convex $\underline{u}\in C^{4}(\Omega)\cap
C^{0}(\overline{\Omega})$ satisfying (1.3) and a locally convex solution $v\in
C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ satisfying (1.6). Then there
exists a strictly locally convex solution $u\in C^{\infty}(\Omega)\cap
C^{0}(\overline{\Omega})$ with $u\geq\underline{u}$ to asymptotic problem
(1.1).

We note that, in order to conduct the Pogorelov type estimate, we have assumed
the existence of $v$ (c.f. [15]). Such $v$ exists if $\Gamma$ bounds a
uniformly convex domain. For the special case $k=n=2$, we do not need to
assume the existence of $v$ (see Theorem 1.2 in [13]) since we have obtained
the pure interior $C^{2}$ estimate.

For general $k$, as known from Sheng-Urbas-Wang [12], Pogorelov type interior
second order estimate is limited to Dirichlet problems with affine boundary
values in Euclidean space. As we know, the affine boundary value spans a
hyperplane, the principal curvatures of which are all zero, and hence satisfy
(1.6) in Euclidean space. The counterpart of the hyperplane in hyperbolic
space is

$\Big{\\{}(x,\sqrt{R^{2}-|x|^{2}})\big{|}\,|x|<R\Big{\\}},$

which intersects $\partial_{\infty}\mathbb{H}^{n+1}$ at
$\Gamma=\\{x\in\mathbb{R}^{n}\big{|}\,|x|=R\\}$, with $R$ a positive constant.
For this reason, we shall restrict our attention to circular asymptotic
boundaries. We shall derive the uniform interior second order estimate (3.27)
by extending the estimate in [12] to hyperbolic space. Our test function and
the subsequent derivations are a little different from [12] due to the
hyperbolic space and the problem setting. As a result, we obtain the following
existence result over disks for general $k$.

######  Theorem 1.2.

Let $k<n$ and $\Omega=\\{x\in\mathbb{R}^{n}||x|<R\\}$. Suppose that
$0<\psi(x,u)\in C^{\infty}(\mathbb{H}^{n+1})$ satisfies

(1.7) $\psi_{u}-\frac{\psi}{u}\geq 0,$

there exists an admissible $\underline{u}\in C^{4}(\Omega)\cap
C^{0}(\overline{\Omega})$ satisfying (1.3) and

(1.8) $-\lambda(D^{2}\underline{u})\in\Gamma_{k+1}\quad\mbox{near}\,\,\Gamma,$

and the compatibility conditions hold for sufficiently small $\epsilon>0$.
Then there exists a unique admissible solution $u\in C^{\infty}(\Omega)\cap
C^{0}(\overline{\Omega})$ with $u\geq\underline{u}$ to the asymptotic problem
(1.1).

We point out that condition (1.7), (1.8) and the compatibility conditions (see
the introduction of [14]) are assumed to guarantee the existence of smooth
admissible solution $u^{\epsilon}\geq\underline{u}$ to (1.4) on
$\overline{\Omega_{\epsilon}}$ (see Theorem 1.7 in [14]). Moreover, (1.7) is
used to derive interior gradient estimate, which assures (1.5). It should also
be emphasized that the assumption $\Omega=\\{x\in\mathbb{R}^{n}||x|<R\\}$ is
assumed solely for Pogorelov type second order interior estimate. In the end,
we provide an example (see also [14]) satisfying the assumptions in our
theorems.

######  Example 1.3.

Let

$\Omega=\big{\\{}x\in\mathbb{R}^{n}\big{|}\,|x|<(1-\sigma^{2})^{\frac{1}{2}}R_{1}\big{\\}},$

where $\sigma\in(0,1)$ and $R_{1}>0$ are constants. Let $\psi=\alpha u^{2}$
with

$\alpha=\frac{\sigma_{k}^{\frac{1}{k}}(\sigma,\ldots,\sigma)}{(1-\sigma)^{2}R_{1}^{2}}.$

Then we may choose $\underline{u}=\sqrt{R_{1}^{2}-|x|^{2}}-\sigma R_{1}$.

This paper is organized as follows: we prove Theorem 1.1 in section 2 and
Theorem 1.2 in section 3.

Acknowledgements The authors would like to thank the referees for giving us
many valuable comments and suggestions.

##  2\. Prescribed Gauss curvature equations

In this section, we shall focus on prescribed Gauss curvature equation, that
is, equation (1.1) with $k=n$. Before we discuss this type of equation, we
first present some preliminary formulae on vertical graph of $u$, which can be
found in [8, 6, 7, 9, 13, 14]. The counterpart in Euclidean space can be found
in [3].

###  2.1. Geometric quantities on vertical graph of $u$

For convenience, denote

$\Sigma=\\{(x,u(x))|\,x\in\Omega\\}.$

The coordinate vector fields on $\Sigma$ are given by

$\partial_{i}+u_{i}\partial_{n+1},\quad i=1,\ldots,n,$

where $\partial_{i}=\frac{\partial}{\partial x^{i}}$ with $i=1,\ldots,n+1$ are
the coordinate vector fields in $\mathbb{R}^{n+1}$.

When $\Sigma$ is considered as a hypersurface in $\mathbb{R}^{n+1}$, the
upward unit normal, metric, inverse of the metric and second fundamental form
are given respectively by

$\nu=\frac{1}{w}(-Du,1),\quad w=\sqrt{1+|Du|^{2}},$
$\tilde{g}_{ij}=\delta_{ij}+u_{i}u_{j},\quad\tilde{g}^{ij}=\delta_{ij}-\frac{u_{i}u_{j}}{w^{2}},\quad\tilde{h}_{ij}=\frac{u_{ij}}{w}.$

The Euclidean principal curvatures $\tilde{\kappa}$ are the eigenvalues of the
symmetric matrix

$\tilde{a}_{ij}=\frac{1}{w}\gamma^{ik}u_{kl}\gamma^{lj},$

where

$\gamma^{ik}=\delta_{ik}-\frac{u_{i}u_{k}}{w(1+w)},\quad\gamma_{ik}=\delta_{ik}+\frac{u_{i}u_{k}}{1+w},$

and

$\gamma^{ik}\gamma_{kj}=\delta_{ij},\quad\gamma_{ik}\gamma_{kj}=\tilde{g}_{ij}.$

When $\Sigma$ is viewed as a hypersurface in $\mathbb{H}^{n+1}$, its unit
upward normal, metric, second fundamental form are given respectively by

${\bf n}=u\nu,\quad g_{ij}=\frac{1}{u^{2}}(\delta_{ij}+u_{i}u_{j}),\quad
h_{ij}=\frac{1}{u^{2}w}(\delta_{ij}+u_{i}u_{j}+uu_{ij}).$

The hyperbolic principal curvatures $\kappa[u]$ are the eigenvalues of the
symmetric matrix $A[u]=\\{a_{ij}\\}$, where

$a_{ij}=u^{2}\gamma^{ik}h_{kl}\gamma^{lj}=\frac{1}{w}\gamma^{ik}(\delta_{kl}+u_{k}u_{l}+uu_{kl})\gamma^{lj}=\frac{1}{w}(\delta_{ij}+u\gamma^{ik}u_{kl}\gamma^{lj}).$

Equation (1.1) can be written as

$f(\kappa[u])=f(\lambda(A[u]))=F(A[u])=\psi(x,u).$

From the above formulae, we observe the following relations.

(2.1)
$h_{ij}=\frac{1}{u}\tilde{h}_{ij}+\frac{\nu^{n+1}}{u^{2}}\tilde{g}_{ij},$

where $\nu^{n+1}=\nu\cdot\partial_{n+1}$ and $\cdot$ is the inner product in
$\mathbb{R}^{n+1}$. Note that formula (2.1) indeed holds for any local frame
on any hypersurface $\Sigma$ which may not be a graph (then
$u=X\cdot\partial_{n+1}$ where $X$ is the position vector of $\Sigma$ in
$\mathbb{R}^{n+1}$). We also have the relation

(2.2) $\kappa_{i}=u\tilde{\kappa_{i}}+\nu^{n+1},\quad i=1,\ldots,n.$

###  2.2. Pogorelov type interior second order estimate

Now we consider the following prescribed Gauss curvature equation

(2.3) $\left\\{\begin{aligned} \det
A[u]=&\psi(x,u,Du)\quad&\mbox{in}\,\,\Omega,\\\
u=&0\quad&\mbox{on}\,\,\Gamma,\end{aligned}\right.$

where $\psi$ is a smooth positive function defined on
$\Omega\times(0,\infty)\times\mathbb{R}^{n}$.

Let $v\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ be a locally convex
solution to the homogeneous equation

(2.4) $\left\\{\begin{aligned}
\det(\delta_{kl}+v_{k}v_{l}+vv_{kl})=&0\quad\mbox{in}\,\,\Omega,\\\
v=&0\quad\mbox{on}\,\,\Gamma.\end{aligned}\right.$

By the transformation $V=v^{2}+|x|^{2}$, one can see that (2.4) is equivalent
to

(2.5) $\left\\{\begin{aligned} \det(V_{kl})=&0\quad&\mbox{in}\,\,\Omega,\\\
V=&|x|^{2}\quad&\mbox{on}\,\,\Gamma.\end{aligned}\right.$

Assume that there exists a strictly locally convex $\underline{u}\in
C^{4}(\Omega)\cap C^{0}(\overline{\Omega})$ such that

(2.6) $\left\\{\begin{aligned} \det
A[\underline{u}]\geq&\psi(x,\underline{u},D\underline{u})\quad&\mbox{in}\,\,\Omega,\\\
\underline{u}=&0\quad&\mbox{on}\,\,\Gamma.\end{aligned}\right.$

Denoting $\underline{U}={\underline{u}}^{2}+|x|^{2}$, we can verify that
$\underline{U}$ is strictly locally convex and satisfies

(2.7) $\left\\{\begin{aligned}
\det(\underline{U}_{kl})\geq&\Psi(x,\underline{U},D\underline{U})\quad&\mbox{in}\,\,\Omega,\\\
\underline{U}=&|x|^{2}\quad&\mbox{on}\,\,\Gamma,\end{aligned}\right.$

where

$\Psi(x,U,DU)=2^{n}\Big{(}\frac{|DU|^{2}-4x\cdot
DU+4U}{4U-4|x|^{2}}\Big{)}^{\frac{n+2}{2}}\psi\Big{(}x,\sqrt{U-|x|^{2}},\frac{DU-2x}{2\sqrt{U-|x|^{2}}}\Big{)}.$

By (2.5), (2.7) and the maximum principle, we know that

$\underline{U}<V\quad\mbox{in}\,\,\Omega.$

By Theorem 1.1 in [13], for $\epsilon>0$ sufficiently small, there exists a
strictly locally convex smooth solution $u^{\epsilon}\geq\underline{u}$ to the
approximating Dirichlet problem

(2.8) $\left\\{\begin{aligned} \det
A[u]=&\psi(x,u,Du)\quad&\mbox{in}\,\,\Omega_{\epsilon},\\\
u=&\epsilon\quad&\mbox{on}\,\,\Gamma_{\epsilon}.\end{aligned}\right.$

Equivalently, we can verify that $U^{\epsilon}=(u^{\epsilon})^{2}+|x|^{2}$ is
a strictly locally convex solution to the Monge-Ampère equation

(2.9) $\left\\{\begin{aligned}
\det(U_{kl})=&\Psi(x,U,DU)\quad&\mbox{in}\,\,\Omega_{\epsilon},\\\
U=&|x|^{2}+\epsilon^{2}=\underline{U}\quad&\mbox{on}\,\,\Gamma_{\epsilon},\end{aligned}\right.$

and $\underline{U}\leq U^{\epsilon}<V$ in $\overline{\Omega_{\epsilon}}$.

For fixed $\epsilon_{0}>0$ sufficiently small, let
$r=d(\Omega_{\epsilon_{0}},\Gamma_{\epsilon_{0}/2})$. For any
$x_{0}\in\Omega_{\epsilon_{0}}$ and for $\tau>0$, consider

$\phi(x)=-\tau(r^{2}-|x-x_{0}|^{2}).$

We note that on $B_{r}(x_{0})$,

$\det(D^{2}V+D^{2}\phi)\leq
C_{1}(n)\sum\limits_{k=1}^{n}\tau^{k}M^{n-k},\quad\mbox{where}\,\,M=\sup_{\Omega_{\epsilon_{0}/2}}|D^{2}V|.$

Also, we observe that on $\overline{\Omega_{\epsilon_{0}/2}}$, for any
$0<\epsilon<\frac{\epsilon_{0}}{4}$,

$C\geq
U^{\epsilon}\geq\underline{U}\geq\frac{\epsilon_{0}^{2}}{4}+|x|^{2}\quad\mbox{and}\quad|DU^{\epsilon}|\leq
C.$

Thus, for any $0<\epsilon<\frac{\epsilon_{0}}{4}$, we have

$\inf_{\Omega_{\epsilon_{0}/2}}\Psi(x,U^{\epsilon},DU^{\epsilon})\geq
C^{-1}>0.$

Now we can choose $\tau$ sufficiently small (depending on $\epsilon_{0}$) such
that for any $x_{0}\in\Omega_{\epsilon_{0}}$, we have on $B_{r}(x_{0})$,

$\det(D^{2}V+D^{2}\phi)\leq\det
D^{2}U^{\epsilon},\quad\forall\,\,0<\epsilon<\frac{\epsilon_{0}}{4}.$

Since $V+\phi=V>U^{\epsilon}$ on $\partial B_{r}(x_{0})$, by the maximum
principle, $V+\phi\geq U^{\epsilon}$ over $B_{r}(x_{0})$. In particular, we
have

$V(x_{0})-\tau r^{2}=(V+\phi)(x_{0})\geq U^{\epsilon}(x_{0}).$

Thus,

(2.10) $\inf_{\Omega_{\epsilon_{0}}}(V-U^{\epsilon})\geq\tau
r^{2},\quad\forall\,\,0<\epsilon<\frac{\epsilon_{0}}{4}.$

Now, set

$\Omega^{\epsilon}_{\epsilon_{0}}=\\{x\in\Omega_{\epsilon}\,\big{|}\,(V-U^{\epsilon})(x)>\frac{1}{2}\tau
r^{2}\\}\quad\mbox{for}\,\,0<\epsilon<\frac{\epsilon_{0}}{4},$

and

$\hat{\Omega}_{\epsilon_{0}}=\\{x\in\Omega\,\big{|}\,(V-\underline{U})(x)\geq\frac{1}{2}\tau
r^{2}\\}.$

It is easy to verify that

(2.11)
$\Omega_{\epsilon_{0}}\subset\subset\Omega^{\epsilon}_{\epsilon_{0}}\subset\hat{\Omega}_{\epsilon_{0}}\subset\subset\Omega,\quad\forall\,\,0<\epsilon<\frac{\epsilon_{0}}{4}.$

Denote

$\delta_{\epsilon_{0}}=\min\limits_{\hat{\Omega}_{\epsilon_{0}}}\underline{u}>0.$

Then, for any $0<\epsilon<\min\\{\epsilon_{0}/4,\delta_{\epsilon_{0}}/2\\}$,
we have

$|x|^{2}+(\delta_{\epsilon_{0}})^{2}\leq U^{\epsilon}\leq
C\quad\mbox{and}\quad|DU^{\epsilon}|\leq
C\quad\mbox{on}\,\,\Omega^{\epsilon}_{\epsilon_{0}},$

and

$V-U^{\epsilon}=\frac{1}{2}\tau
r^{2}\quad\mbox{on}\,\,\partial\Omega^{\epsilon}_{\epsilon_{0}}.$

Now, we apply Trudinger-Urbas’ method [15] to obtain second order Pogorelov
type interior estimate. Consider in $\Omega_{\epsilon_{0}}^{\epsilon}$,

$\ln\eta+\frac{\beta}{2}|DU^{\epsilon}|^{2}+\ln
U^{\epsilon}_{\xi\xi},\quad\mbox{where}\,\,\eta=V-U^{\epsilon}-\frac{1}{2}\tau
r^{2}.$

The maximum value must be achieved at an interior point in
$\Omega_{\epsilon_{0}}^{\epsilon}$. We assume it to be $0$ and in direction
$\xi={\partial}_{1}$. In addition, we assume that
$\big{(}U^{\epsilon}_{ij}(0)\big{)}$ is diagonal after we rotate
$\partial_{2},\ldots,\partial_{n}$. For convenience, we omit the superscript
$\epsilon$ in $U^{\epsilon}$. Thus, the function

$\ln\eta+\frac{\beta}{2}|DU|^{2}+\ln U_{11}$

achieves its maximum at $0$, at which, we have

(2.12) $\frac{\eta_{i}}{\eta}+\beta U_{i}U_{ii}+\frac{U_{11i}}{U_{11}}=0,$
(2.13)
$\frac{U^{ij}\eta_{ij}}{\eta}-\frac{U^{ij}\eta_{i}\eta_{j}}{\eta^{2}}+\beta
U_{k}U^{ij}U_{kij}+\beta\Delta
U+\frac{U^{ij}U_{11ij}}{U_{11}}-\frac{U^{ij}U_{11i}U_{11j}}{U_{11}^{2}}\leq
0,$

where $U^{ij}U_{jk}=\delta^{i}_{k}$. Since $U$ is strictly locally convex, it
suffices to give an upper bound for $\eta U_{11}$. We may assume that
$U_{11}\geq 1$. Write equation (2.9) as

$\ln\det(D^{2}U)=\ln\Psi(x,U,DU)=\tilde{\Psi}(x,U,DU).$

Differentiate this equation, we obtain

(2.14)
$U^{ij}U_{ijk}=\tilde{\Psi}_{x_{k}}+\tilde{\Psi}_{U}U_{k}+\tilde{\Psi}_{U_{i}}U_{ik},$
(2.15)
$U^{ij}U_{ij11}-U^{ik}U^{jl}U_{ij1}U_{kl1}\geq\tilde{\Psi}_{U_{i}}U_{i11}-CU_{11}^{2}.$

Also, we have

(2.16) $U^{ij}\eta_{ij}=U^{ij}(V_{ij}-U_{ij})\geq-n.$

Taking (2.12) and (2.14)–(2.16) into (2.13) yields,

(2.17) $-\frac{U^{ii}\eta_{i}^{2}}{\eta^{2}}-C\beta+\beta
U_{11}+\frac{U^{ii}U^{jj}U_{ij1}^{2}}{U_{11}}-\frac{C}{\eta}-CU_{11}-\frac{U^{ii}U_{11i}^{2}}{U_{11}^{2}}\leq
0.$

Assuming $\eta U_{11}\geq 1$, again by (2.12), we have

$\displaystyle\frac{U^{ii}\eta_{i}^{2}}{\eta^{2}}=$
$\displaystyle\frac{U^{11}\eta_{1}^{2}}{\eta^{2}}+\sum_{i\neq
1}U^{ii}\Big{(}\beta U_{i}U_{ii}+\frac{U_{11i}}{U_{11}}\Big{)}^{2}$
$\displaystyle=$ $\displaystyle\frac{U^{11}\eta_{1}^{2}}{\eta^{2}}+\sum_{i\neq
1}\Big{(}\beta^{2}U_{i}^{2}U_{ii}+2\beta
U_{i}\big{(}-\frac{\eta_{i}}{\eta}-\beta
U_{i}U_{ii}\big{)}+U^{ii}\frac{U_{11i}^{2}}{U_{11}^{2}}\Big{)}$
$\displaystyle\leq$
$\displaystyle\frac{C}{\eta}+\frac{C\beta}{\eta}+\sum_{i\neq
1}\frac{U^{ii}U_{11i}^{2}}{U_{11}^{2}}.$

Hence, (2.17) reduces to

(2.18)
$\frac{U^{ii}U^{jj}U_{ij1}^{2}}{U_{11}}-\frac{U^{ii}U_{11i}^{2}}{U_{11}^{2}}-\sum\limits_{i\neq
1}\frac{U^{ii}U_{11i}^{2}}{U_{11}^{2}}+\beta
U_{11}-CU_{11}-\frac{C}{\eta}-\frac{C\beta}{\eta}\leq 0.$

Also,

$\displaystyle\frac{U^{ii}U^{jj}U_{ij1}^{2}}{U_{11}}=$
$\displaystyle\frac{U^{ii}U^{11}U_{i11}^{2}}{U_{11}}+\sum\limits_{j\neq
1}\frac{U^{11}U^{jj}U_{1j1}^{2}}{U_{11}}+\sum\limits_{j\neq 1,i\neq
1}\frac{U^{ii}U^{jj}U_{ij1}^{2}}{U_{11}}$ $\displaystyle\geq$
$\displaystyle\frac{U^{ii}U_{11i}^{2}}{U_{11}^{2}}+\sum\limits_{j\neq
1}\frac{U^{jj}U_{11j}^{2}}{U_{11}^{2}}.$

Hence (2.18) becomes

$\beta U_{11}-CU_{11}-\frac{C}{\eta}-\frac{C\beta}{\eta}\leq 0.$

Choosing $\beta$ sufficiently large, we thus obtain an upper bound $\eta
U_{11}\leq C$. Therefore, by (2.10), we arrive at

(2.19) $\sup_{\Omega_{\epsilon_{0}}}|D^{2}U^{\epsilon}|\leq
C,\quad\forall\,\,0<\epsilon<\min\\{\epsilon_{0}/4,\delta_{\epsilon_{0}}/2\\}.$

##  3\. Prescribed $k$th Weingarten curvature equations

In this section, we consider prescribed $k$th Weingarten curvature equation

(3.1) $\left\\{\begin{aligned}
f(\kappa[u])=&F(A[u])=\sigma_{k}^{1/k}(A[u])=\psi(x,u)\quad&\mbox{in}\,\,\Omega,\\\
u=&0\quad&\mbox{on}\,\,\Gamma.\end{aligned}\right.$

Motivated by the idea of Trudinger-Urbas [15], we assume that there exists a
solution $v\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ with
$\kappa[v]\in\overline{\Gamma_{k}}$ to the homogeneous equation

(3.2) $\left\\{\begin{aligned} f(\kappa[v])=&0\quad\mbox{in}\,\,\Omega,\\\
v=&0\quad\mbox{on}\,\,\Gamma.\end{aligned}\right.$

Since $\kappa[\underline{u}]\in\Gamma_{k}$ and
$\kappa[v]\in\partial\Gamma_{k}$, we have

$\underline{u}<v\quad\mbox{in}\,\,\Omega.$

In fact, suppose $\underline{u}-v$ achieves a nonnegative maximum at
$x_{0}\in\Omega$. Then we have

$\underline{u}(x_{0})\geq v(x_{0}),\quad D\underline{u}(x_{0})=Dv(x_{0}),\quad
D^{2}\underline{u}(x_{0})\leq D^{2}v(x_{0}).$

But then at $x_{0}$,

$\delta_{ij}+v{\gamma}^{ik}v_{kl}\gamma^{lj}\geq\delta_{ij}+v{\gamma}^{ik}\underline{u}_{kl}\gamma^{lj}=\Big{(}1-\frac{v}{\underline{u}}\Big{)}\delta_{ij}+\frac{v}{\underline{u}}\Big{(}\delta_{ij}+\underline{u}\gamma^{ik}\underline{u}_{kl}\gamma^{lj}\Big{)},$

which leads to a contradiction.

For $\epsilon>0$ sufficiently small, let $u^{\epsilon}\geq\underline{u}$ be a
smooth admissible solution to the approximating Dirichlet problem (1.4).
Again, we have $\underline{u}\leq u^{\epsilon}<v$ in
$\overline{\Omega_{\epsilon}}$.

For fixed $\epsilon_{0}>0$ sufficiently small, let
$r=d(\Omega_{\epsilon_{0}},\Gamma_{\epsilon_{0}/2})$. For any
$x_{0}\in\Omega_{\epsilon_{0}}$ and for sufficiently small $\tau>0$ (depending
on $\epsilon_{0}$), consider on $B_{r}(x_{0})$,

$v^{\tau}=\sqrt{v^{2}-\tau(r^{2}-|x-x_{0}|^{2})}>0.$

Since

$Dv^{\tau}=Dv+\mathcal{O}(\tau),\quad|Dv^{\tau}|^{2}=|Dv|^{2}+\mathcal{O}(\tau),$

we have

$\delta_{il}-\frac{v^{\tau}_{i}v^{\tau}_{l}}{1+|Dv^{\tau}|^{2}}=\delta_{il}-\frac{v_{i}v_{l}}{1+|Dv|^{2}}+\mathcal{O}(\tau).$

Also,

$\delta_{ij}+v^{\tau}_{i}v^{\tau}_{j}+v^{\tau}v^{\tau}_{ij}=\frac{1}{2}\Big{(}v^{2}+|x|^{2}-\tau(r^{2}-|x-x_{0}|^{2})\Big{)}_{ij}=\delta_{ij}+v_{i}v_{j}+vv_{ij}+\tau\delta_{ij}.$

Therefore, we have

(3.3)
$\displaystyle\sigma_{k}\Big{(}\big{(}\delta_{il}-\frac{v^{\tau}_{i}v^{\tau}_{l}}{1+|Dv^{\tau}|^{2}}\big{)}\big{(}\delta_{lj}+v^{\tau}_{l}v^{\tau}_{j}+v^{\tau}v^{\tau}_{lj}\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\sigma_{k}\Big{(}\big{(}\delta_{il}-\frac{v_{i}v_{l}}{1+|Dv|^{2}}\big{)}\big{(}\delta_{lj}+v_{l}v_{j}+vv_{lj}\big{)}+\mathcal{O}(\tau)\Big{)}$
$\displaystyle\leq$
$\displaystyle\sigma_{k}\Big{(}\big{(}\delta_{il}-\frac{v_{i}v_{l}}{1+|Dv|^{2}}\big{)}\big{(}\delta_{lj}+v_{l}v_{j}+vv_{lj}\big{)}\Big{)}+C\tau=C\tau.$

Here $C$ in the last row of (3.3) depends on $n$, $k$, $\epsilon_{0}$ and
$\|v\|_{C^{2}(\Omega_{\epsilon_{0}/2})}$. The argument is similar as in
section 2.

In [14], we have obtained the $C^{0}$ estimate

$\epsilon\leq\underline{u}\leq u^{\epsilon}\leq C_{0}\quad\text{on
}\overline{\Omega_{\epsilon}},$

where $C_{0}$ is independent of $\epsilon$. Therefore we have

$\inf_{\Omega_{\epsilon_{0}/2}}\psi(x,u^{\epsilon})\geq
C^{-1}>0,\quad\forall\,\,0<\epsilon<\frac{\epsilon_{0}}{2}.$

By (3.3), we can choose $\tau$ sufficiently small (depending on
$\epsilon_{0}$) such that for any $x_{0}\in\Omega_{\epsilon_{0}}$, we have on
$B_{r}(x_{0})$,

(3.4)
$\sigma_{k}(A[v^{\tau}])<\frac{(v^{\tau})^{k}}{(u^{\epsilon})^{k}}\sigma_{k}(A[u^{\epsilon}]),\quad\forall\,\,0<\epsilon<\frac{\epsilon_{0}}{2}.$

Since $v^{\tau}=v>u^{\epsilon}$ on $\partial B_{r}(x_{0})$, we must have
$v^{\tau}>u^{\epsilon}$ over $B_{r}(x_{0})$. In fact, suppose
$v^{\tau}-u^{\epsilon}$ achieves a nonpositive minimum at $x_{1}\in
B_{r}(x_{0})$. Then we have

$v^{\tau}(x_{1})\leq u^{\epsilon}(x_{1}),\quad
Dv^{\tau}(x_{1})=Du^{\epsilon}(x_{1}),\quad D^{2}v^{\tau}(x_{1})\geq
D^{2}u^{\epsilon}(x_{1}).$

But then at $x_{1}$,

$\displaystyle\sigma_{k}\Big{(}\frac{A[v^{\tau}]}{v^{\tau}}\Big{)}=\sigma_{k}\Big{(}\frac{1}{w}\big{(}\frac{\delta_{ij}}{v^{\tau}}+\gamma^{il}v^{\tau}_{lm}\gamma^{mj}\big{)}\Big{)}$
$\displaystyle\geq$
$\displaystyle\sigma_{k}\Big{(}\frac{1}{w}\big{(}\frac{\delta_{ij}}{u^{\epsilon}}+\gamma^{il}u^{\epsilon}_{lm}\gamma^{mj}\big{)}\Big{)}=\sigma_{k}\Big{(}\frac{A[u^{\epsilon}]}{u^{\epsilon}}\Big{)},$

contradicting (3.4). In particular, we have

$v^{2}(x_{0})-\tau r^{2}>(u^{\epsilon})^{2}(x_{0}),$

which implies that

(3.5) $\inf_{\Omega_{\epsilon_{0}}}\big{(}v^{2}-(u^{\epsilon})^{2}\big{)}\geq
2c:=\tau r^{2}>0,\quad\forall\,\,0<\epsilon<\frac{\epsilon_{0}}{2}.$

Note that $c$ depends on $\epsilon_{0}$ but is independent of $\epsilon$.

Similar to last section, define for $0<\epsilon<\frac{\epsilon_{0}}{2}$,

$\Omega^{\epsilon}_{\epsilon_{0}}=\\{x\in\Omega_{\epsilon}\,\big{|}\,\big{(}v^{2}-(u^{\epsilon})^{2}\big{)}(x)>c\\}\quad\mbox{and}\quad\hat{\Omega}_{\epsilon_{0}}=\\{x\in\Omega\,\big{|}\,(v^{2}-\underline{u}^{2})(x)\geq
c\\}.$

We can verify that

(3.6)
$\Omega_{\epsilon_{0}}\subset\subset\Omega^{\epsilon}_{\epsilon_{0}}\subset\hat{\Omega}_{\epsilon_{0}}\subset\subset\Omega,\quad\forall\,\,0<\epsilon<\frac{\epsilon_{0}}{2}.$

Denote

$\delta_{\epsilon_{0}}=\min\limits_{\hat{\Omega}_{\epsilon_{0}}}\underline{u}>0.$

We know that for any
$0<\epsilon<\min\\{\epsilon_{0}/2,\delta_{\epsilon_{0}}/2\\}=\delta_{\epsilon_{0}}/2$,

$\delta_{\epsilon_{0}}\leq u^{\epsilon}\leq C_{0},\quad|Du^{\epsilon}|\leq
C\quad\mbox{on}\,\,\Omega^{\epsilon}_{\epsilon_{0}}$

and

$v^{2}-(u^{\epsilon})^{2}=c\quad\mbox{on}\,\,\partial\Omega^{\epsilon}_{\epsilon_{0}}.$

Now we want to establish Pogorelov type interior curvature estimate for the
graph of $u^{\epsilon}$ on $\Omega_{\epsilon_{0}}^{\epsilon}$ for any
$0<\epsilon<\delta_{\epsilon_{0}}/2$, i.e.

(3.7) $\max_{1\leq i\leq
n}\big{|}\kappa_{i}[u^{\epsilon}](x)\big{|}\,\leq\,\frac{C}{\big{(}v^{2}-(u^{\epsilon})^{2}-c\big{)}^{b}},\quad\forall\,\,x\in\Omega_{\epsilon_{0}}^{\epsilon},\quad\forall\,\,0<\epsilon<\frac{\delta_{\epsilon_{0}}}{2},$

where $b\geq 1$ is a constant to be determined later. For convenience, we omit
the superscript in $u^{\epsilon}$ and denote the graph of $u$ over
$\Omega_{\epsilon_{0}}^{\epsilon}$ by $\Sigma$.

At this point, we assume that $v=\sqrt{R^{2}-|x|^{2}}$. Consequently, $\Omega$
has to be

$\Omega=\\{x\in\mathbb{R}^{n}|\,|x|<R\\}.$

Note that $\kappa[v]=(0,\ldots,0)$, which can be computed easily by formula
(2.2). Hence $v$ satisfies (3.2).

In order to perform calculations on $\Sigma$, we may need the following
preliminary formulae (see [7, 13, 14]).

###  3.1. Calculations on $\Sigma$

For a hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$, let $g$ and $\nabla$ be the
induced metric and Levi-Civita connection on $\Sigma$. Let $\tilde{g}$ and
$\tilde{\nabla}$ be the metric and Levi-Civita connection on $\Sigma$ induced
from $\mathbb{R}^{n+1}$. Let $X$ be the position vector of $\Sigma$ in
$\mathbb{R}^{n+1}$ and set $u=X\cdot\partial_{n+1}$. Since the associated
Christoffel symbols have the following relation

$\Gamma_{ij}^{k}=\tilde{\Gamma}_{ij}^{k}-\frac{1}{u}(u_{i}\delta_{kj}+u_{j}\delta_{ik}-\tilde{g}^{kl}u_{l}\tilde{g}_{ij}),$

we can prove that for any $w\in C^{2}(\Sigma)$ and any local frame on
$\Sigma$,

(3.8)
$\nabla_{ij}w=(w_{i})_{j}-\Gamma_{ij}^{k}w_{k}=\tilde{\nabla}_{ij}w+\frac{1}{u}(u_{i}w_{j}+u_{j}w_{i}-\tilde{g}^{kl}u_{l}w_{k}\tilde{g}_{ij}).$

######  Lemma 3.1.

When $\Sigma$ is viewed as a hypersurface in $\mathbb{R}^{n+1}$, we have

$u_{i}=\tau_{i}\cdot\partial_{n+1},\quad(x^{\alpha})_{i}=\tau_{i}\cdot\partial_{\alpha},\quad\alpha=1,\ldots,n,$

$\tilde{g}^{kl}u_{k}u_{l}=|\tilde{\nabla}u|^{2}=1-(\nu^{n+1})^{2},$

$\tilde{\nabla}_{ij}u=\tilde{h}_{ij}\nu^{n+1},\quad\tilde{\nabla}_{ij}x^{\alpha}=\tilde{h}_{ij}\nu^{\alpha},\quad\alpha=1,\ldots,n,$

$(\nu^{n+1})_{i}=-\tilde{h}_{ij}\,\tilde{g}^{jk}u_{k},$

and

$\tilde{\nabla}_{ij}\nu^{n+1}=-\tilde{g}^{kl}(\nu^{n+1}\tilde{h}_{il}\tilde{h}_{kj}+u_{l}\tilde{\nabla}_{k}\tilde{h}_{ij}),$

where $\tau_{1},\ldots,\tau_{n}$ is any local frame on $\Sigma$,
$X=(x^{1},\ldots,x^{n},u)$ is the position vector field on $\Sigma$ and
$\nu=(\nu^{1},\ldots,\nu^{n+1})$ is the unit normal vector field to $\Sigma$
in $\mathbb{R}^{n+1}$.

###### Proof.

The proof can be found in [13]. ∎

For convenience, denote

$f_{i}=\frac{\partial f}{\partial\kappa_{i}}\quad\text{and}\quad
F^{ij}=\frac{\partial F}{\partial a_{ij}}.$

######  Lemma 3.2.

Let $\Sigma$ be an admissible hypersurface in $\mathbb{H}^{n+1}$ satisfying
$f(\kappa)=\psi$. Then in a local orthonormal frame on $\Sigma$, we have

$\displaystyle F^{ij}\nabla_{ij}\nu^{n+1}=$
$\displaystyle-\nu^{n+1}F^{ij}h_{ik}h_{kj}+\big{(}1+(\nu^{n+1})^{2}\big{)}F^{ij}h_{ij}-\nu^{n+1}\sum
f_{i}$
$\displaystyle-\frac{2}{u^{2}}F^{ij}h_{jk}u_{i}u_{k}+\frac{2\nu^{n+1}}{u^{2}}F^{ij}u_{i}u_{j}-\frac{u_{k}}{u}\psi_{k}.$

###### Proof.

The proof can be found in [13]. ∎

###  3.2. Pogorelov type interior curvature estimate

Note that on $\Omega_{\epsilon_{0}}^{\epsilon}$, in view of (3.6), we have

$\nu^{n+1}=\frac{1}{\sqrt{1+|Du|^{2}}}\geq 2a>0,$

where $a$ is a positive constant depending on $\epsilon_{0}$ but independent
of $\epsilon$. Let $\kappa_{\max}(X)$ be the largest principal curvature of
$\Sigma$ at $X$. Consider the test function

$M_{0}=\sup\limits_{X\in\Sigma}(v^{2}-u^{2}-c)^{b}\frac{\kappa_{\max}(X)}{{\nu}^{n+1}-a}e^{\frac{\beta}{u}},$

where $b\geq 1$ and $\beta>0$ are constants to be determined. Obviously,
$M_{0}>0$ is attained at an interior point $X_{0}\in\Sigma$.

Let $\tau_{1},\ldots,\tau_{n}$ be a local orthonormal frame with respect to
hyperbolic metric about $X_{0}$ such that
$h_{ij}(X_{0})=\kappa_{i}\delta_{ij}$, where
$\kappa_{1}\geq\ldots\geq\kappa_{n}$ are the principal curvatures of $\Sigma$
at $X_{0}$. Then

$\ln h_{11}-\ln({\nu}^{n+1}-a)+\frac{\beta}{u}+b\ln(v^{2}-u^{2}-c)$

also achieves a local maximum at $X_{0}$, at which, we have

(3.9)
$\frac{h_{11i}}{h_{11}}-\frac{\nabla_{i}\nu^{n+1}}{\nu^{n+1}-a}-\beta\frac{u_{i}}{u^{2}}+b\frac{(v^{2}-u^{2})_{i}}{v^{2}-u^{2}-c}=0,$
(3.10)
$\displaystyle\frac{F^{ii}h_{11ii}}{h_{11}}-\frac{F^{ii}h_{11i}^{2}}{h_{11}^{2}}-\frac{F^{ii}\nabla_{ii}\nu^{n+1}}{\nu^{n+1}-a}+\frac{F^{ii}(\nu^{n+1})_{i}^{2}}{(\nu^{n+1}-a)^{2}}$
$\displaystyle-\beta\frac{F^{ii}\nabla_{ii}u}{u^{2}}+\beta
F^{ii}\frac{2u_{i}^{2}}{u^{3}}+b\frac{F^{ii}\nabla_{ii}(v^{2}-u^{2})}{v^{2}-u^{2}-c}-bF^{ii}\frac{(v^{2}-u^{2})_{i}^{2}}{(v^{2}-u^{2}-c)^{2}}\leq
0.$

Note that $v^{2}=R^{2}-|x|^{2}$. By (3.8), Lemma 3.1 and Lemma 3.2, we have

(3.11) $F^{ii}\nabla_{ii}u=u\nu^{n+1}\psi+\frac{2}{u}\sum f_{i}u_{i}^{2}-u\sum
f_{i},$ (3.12) $\displaystyle
F^{ii}\nabla_{ii}(v^{2}-u^{2})=F^{ii}\nabla_{ii}(R^{2}-|X|^{2})$
$\displaystyle=$ $\displaystyle
F^{ii}\Big{(}-\tilde{\nabla}_{ii}|X|^{2}-\frac{2u_{i}}{u}(|X|^{2})_{i}+\sum_{k=1}^{n}\frac{u_{k}}{u}(|X|^{2})_{k}\Big{)}$
$\displaystyle=$ $\displaystyle
F^{ii}\Big{(}-(2X\cdot\tau_{i})_{i}+\tilde{\Gamma}_{ii}^{k}2X\cdot\tau_{k}-\frac{2u_{i}}{u}(2X\cdot\tau_{i})+\sum_{k=1}^{n}\frac{u_{k}}{u}(2X\cdot\tau_{k})\Big{)}$
$\displaystyle=$ $\displaystyle
F^{ii}\Big{(}-2u^{2}-2X\cdot\nu\tilde{h}_{ii}-\frac{2u_{i}}{u}(2X\cdot\tau_{i})+\sum_{k=1}^{n}\frac{u_{k}}{u}(2X\cdot\tau_{k})\Big{)}$
$\displaystyle=$ $\displaystyle
F^{ii}\Big{(}-2u^{2}-2X\cdot\nu(\kappa_{i}-\nu^{n+1})u-\frac{2u_{i}}{u}(2X\cdot\tau_{i})+\sum_{k=1}^{n}\frac{u_{k}}{u}(2X\cdot\tau_{k})\Big{)}$
$\displaystyle=$ $\displaystyle
F^{ii}\bigg{(}-2u^{2}+2u(X\cdot\nu)(\nu\cdot\partial_{n+1})-\frac{4u_{i}}{u}(X\cdot\tau_{i})$
$\displaystyle+2u\sum_{k=1}^{n}\Big{(}\frac{\tau_{k}}{u}\cdot\partial_{n+1}\Big{)}\Big{(}X\cdot\frac{\tau_{k}}{u}\Big{)}\bigg{)}-2(X\cdot\nu)u\sum
f_{i}\kappa_{i}$ $\displaystyle=$
$\displaystyle-2(X\cdot\nu)u\psi-4f_{i}\frac{u_{i}}{u}(X\cdot\tau_{i}),$

and

(3.13) $\displaystyle F^{ii}\nabla_{ii}\nu^{n+1}=$
$\displaystyle-\nu^{n+1}\sum
f_{i}\kappa_{i}^{2}+\big{(}1+(\nu^{n+1})^{2}\big{)}\psi-\nu^{n+1}\sum f_{i}$
$\displaystyle-\sum\frac{2f_{i}\kappa_{i}u_{i}^{2}}{u^{2}}+\sum\frac{2\nu^{n+1}f_{i}u_{i}^{2}}{u^{2}}-\frac{u_{k}}{u}\psi_{k}.$

Also, by Gauss equation, we have the following commutation formula,

(3.14) $h_{ii11}=h_{11ii}+(\kappa_{i}\kappa_{1}-1)(\kappa_{i}-\kappa_{1}).$

Differentiating equation (3.1) twice, we obtain

(3.15) $F^{ii}h_{ii11}+F^{ij,rs}h_{ij1}h_{rs1}=\psi_{11}\geq-C\kappa_{1}.$

Taking (3.11)–(3.15) into (3.10) yields,

(3.16)
$\displaystyle\Big{(}\kappa_{1}-\frac{1+(\nu^{n+1})^{2}}{\nu^{n+1}-a}-\frac{\beta\nu^{n+1}}{u}-\frac{2b(X\cdot\nu)u}{v^{2}-u^{2}-c}\Big{)}\psi-C$
$\displaystyle+\Big{(}\frac{\beta}{u}+\frac{a}{\nu^{n+1}-a}\Big{)}\sum
f_{i}+\frac{a}{\nu^{n+1}-a}\sum f_{i}\kappa_{i}^{2}$
$\displaystyle+\frac{2}{\nu^{n+1}-a}\sum
f_{i}\kappa_{i}\frac{u_{i}^{2}}{u^{2}}-\frac{2\nu^{n+1}}{\nu^{n+1}-a}\sum
f_{i}\frac{u_{i}^{2}}{u^{2}}$ $\displaystyle-b\sum
f_{i}\frac{(v^{2}-u^{2})_{i}^{2}}{(v^{2}-u^{2}-c)^{2}}-\frac{4bf_{i}u_{i}(X\cdot\tau_{i})}{u(v^{2}-u^{2}-c)}$
$\displaystyle-\frac{F^{ij,rs}h_{ij1}h_{rs1}}{\kappa_{1}}-\frac{F^{ii}h_{11i}^{2}}{\kappa_{1}^{2}}+\frac{F^{ii}(\nu^{n+1})_{i}^{2}}{(\nu^{n+1}-a)^{2}}\leq
0.$

By Lemma 3.1 and Cauchy-Schwartz inequality, we have

$\displaystyle\frac{2}{\nu^{n+1}-a}\sum
f_{i}\kappa_{i}\frac{u_{i}^{2}}{u^{2}}-\frac{2\nu^{n+1}}{\nu^{n+1}-a}\sum
f_{i}\frac{u_{i}^{2}}{u^{2}}$ $\displaystyle\geq$
$\displaystyle-\frac{2}{\nu^{n+1}-a}\sum
f_{i}|\kappa_{i}|-\frac{2}{\nu^{n+1}-a}\sum f_{i}$ $\displaystyle\geq$
$\displaystyle-\frac{4}{a(\nu^{n+1}-a)}\sum f_{i}-\frac{a}{4(\nu^{n+1}-a)}\sum
f_{i}\kappa_{i}^{2}-\frac{2}{\nu^{n+1}-a}\sum f_{i}.$

Hence, (3.16) reduces to

(3.17)
$\displaystyle\Big{(}\kappa_{1}-\frac{1+(\nu^{n+1})^{2}}{\nu^{n+1}-a}-\frac{\beta\nu^{n+1}}{u}-\frac{2b(X\cdot\nu)u}{v^{2}-u^{2}-c}\Big{)}\psi-C$
$\displaystyle+\Big{(}\frac{\beta}{u}+\frac{a-2-4a^{-1}}{\nu^{n+1}-a}\Big{)}\sum
f_{i}+\frac{3a}{4(\nu^{n+1}-a)}\sum f_{i}\kappa_{i}^{2}$ $\displaystyle-b\sum
f_{i}\frac{(v^{2}-u^{2})_{i}^{2}}{(v^{2}-u^{2}-c)^{2}}-\frac{4bf_{i}u_{i}(X\cdot\tau_{i})}{u(v^{2}-u^{2}-c)}$
$\displaystyle-\frac{F^{ij,rs}h_{ij1}h_{rs1}}{\kappa_{1}}-\frac{F^{ii}h_{11i}^{2}}{\kappa_{1}^{2}}+\frac{F^{ii}(\nu^{n+1})_{i}^{2}}{(\nu^{n+1}-a)^{2}}\leq
0.$

Let $\theta\in(0,1)$ be a constant which will be specified later. We divide
our discussion into two cases.

Case (i). Assume that $\kappa_{n}\leq-\theta\kappa_{1}$. By (3.9) and Cauchy-
Schwartz inequality, we have

(3.18)
$\displaystyle\frac{F^{ii}h_{11i}^{2}}{\kappa_{1}^{2}}\leq(1+\delta_{1})f_{i}\frac{(\nu^{n+1})_{i}^{2}}{(\nu^{n+1}-a)^{2}}$
$\displaystyle+2(1+\delta_{1}^{-1})\beta^{2}f_{i}\frac{u_{i}^{2}}{u^{4}}+2(1+\delta_{1}^{-1})b^{2}f_{i}\frac{(v^{2}-u^{2})_{i}^{2}}{(v^{2}-u^{2}-c)^{2}},$

where $\delta_{1}$ is a positive constant to be determined later. Also, we
have

(3.19)
$-\frac{4bf_{i}u_{i}(X\cdot\tau_{i})}{u(v^{2}-u^{2}-c)}=\frac{2bf_{i}u_{i}(v^{2}-u^{2})_{i}}{u(v^{2}-u^{2}-c)}\geq-\sum
f_{i}-b^{2}f_{i}\frac{(v^{2}-u^{2})_{i}^{2}}{(v^{2}-u^{2}-c)^{2}}.$

Taking (3.18) and (3.19) into (3.17) and noting that
$-F^{ij,rs}h_{ij1}h_{rs1}\geq 0$, we obtain

(3.20)
$\displaystyle\Big{(}\kappa_{1}-\frac{1+(\nu^{n+1})^{2}}{\nu^{n+1}-a}-\frac{\beta\nu^{n+1}}{u}-\frac{2b(X\cdot\nu)u}{v^{2}-u^{2}-c}\Big{)}\psi-C$
$\displaystyle+\Big{(}\frac{\beta}{u}+\frac{a-2-4a^{-1}}{\nu^{n+1}-a}-1-2(1+\delta_{1}^{-1})\frac{\beta^{2}}{u^{2}}-\frac{Cb+C(3+2\delta_{1}^{-1})b^{2}}{(v^{2}-u^{2}-c)^{2}}\Big{)}\sum
f_{i}$ $\displaystyle+\frac{3a}{4(\nu^{n+1}-a)}\sum
f_{i}\kappa_{i}^{2}-\delta_{1}\frac{F^{ii}(\nu^{n+1})_{i}^{2}}{(\nu^{n+1}-a)^{2}}\leq
0.$

By Lemma 3.1 and (2.1), we know that

$(\nu^{n+1})_{i}=\frac{u_{i}}{u}(\nu^{n+1}-\kappa_{i}),$

and consequently,

(3.21)
$\frac{F^{ii}(\nu^{n+1})_{i}^{2}}{(\nu^{n+1}-a)^{2}}\leq\frac{2}{(\nu^{n+1}-a)^{2}}\sum
f_{i}\kappa_{i}^{2}+\frac{2}{(\nu^{n+1}-a)^{2}}\sum f_{i}.$

Taking (3.21) into (3.20) and choosing $\delta_{1}=\frac{a^{2}}{8}$, we have

$\displaystyle\Big{(}\kappa_{1}-\frac{1+(\nu^{n+1})^{2}}{\nu^{n+1}-a}-\frac{\beta\nu^{n+1}}{u}-\frac{2b(X\cdot\nu)u}{v^{2}-u^{2}-c}\Big{)}\psi-C+\frac{a}{2(\nu^{n+1}-a)}\sum
f_{i}\kappa_{i}^{2}$
$\displaystyle+\Big{(}\frac{\beta}{u}+\frac{\frac{3}{4}a-2-\frac{4}{a}}{\nu^{n+1}-a}-1-\frac{2(1+8a^{-2})\beta^{2}}{u^{2}}-C\frac{b+(3+16a^{-2})b^{2}}{(v^{2}-u^{2}-c)^{2}}\Big{)}\sum
f_{i}\leq 0.$

Since

$\sum f_{i}\kappa_{i}^{2}\geq
f_{n}\kappa_{n}^{2}\geq\frac{\theta^{2}}{n}\kappa_{1}^{2}\sum f_{i},$

we obtain an upper bound for $(v^{2}-u^{2}-c)\kappa_{1}$.

Case (ii). Assume that $\kappa_{n}>-\theta\kappa_{1}$. Denote

$I=\\{i\,|\,f_{1}\geq\theta^{2}f_{i}\\},\quad
J=\\{i\,|\,f_{1}<\theta^{2}f_{i}\\}.$

By (3.9) and Cauchy-Schwartz inequality, we have

(3.22) $\displaystyle\sum
f_{i}\frac{(v^{2}-u^{2})_{i}^{2}}{(v^{2}-u^{2}-c)^{2}}\leq\sum_{j\in
I}f_{j}\frac{(v^{2}-u^{2})_{j}^{2}}{(v^{2}-u^{2}-c)^{2}}$
$\displaystyle+\frac{2}{b^{2}}\sum_{j\in
J}f_{j}\Big{(}\frac{(\nu^{n+1})_{j}^{2}}{(\nu^{n+1}-a)^{2}}+\frac{2h_{11j}^{2}}{h_{11}^{2}}+2\beta^{2}\frac{u_{j}^{2}}{u^{4}}\Big{)},$
(3.23)
$\displaystyle\frac{F^{ii}h_{11i}^{2}}{h_{11}^{2}}\leq(1+\delta_{2})\sum_{j\in
I}f_{j}\frac{(\nu^{n+1})_{j}^{2}}{(\nu^{n+1}-a)^{2}}+2(1+\delta_{2}^{-1})\beta^{2}\sum_{j\in
I}f_{j}\frac{u_{j}^{2}}{u^{4}}$
$\displaystyle+2(1+\delta_{2}^{-1})b^{2}\sum_{j\in
I}f_{j}\frac{(v^{2}-u^{2})_{j}^{2}}{(v^{2}-u^{2}-c)^{2}}+\sum_{j\in
J}\frac{F^{jj}h_{11j}^{2}}{h_{11}^{2}},$

and

(3.24)
$\displaystyle-\frac{4bf_{i}u_{i}(X\cdot\tau_{i})}{u(v^{2}-u^{2}-c)}=\frac{2bf_{i}u_{i}(v^{2}-u^{2})_{i}}{u(v^{2}-u^{2}-c)}$
$\displaystyle\geq$ $\displaystyle\sum_{j\in
I}\frac{2bf_{j}u_{j}(v^{2}-u^{2})_{j}}{u(v^{2}-u^{2}-c)}+\sum_{j\in
J}\frac{2f_{j}u_{j}}{u}\Big{(}-\frac{h_{11j}}{h_{11}}+\frac{(\nu^{n+1})_{j}}{\nu^{n+1}-a}\Big{)}$
$\displaystyle\geq$ $\displaystyle-\sum_{j\in I}f_{j}-b^{2}\sum_{j\in
I}f_{j}\frac{(v^{2}-u^{2})_{j}^{2}}{(v^{2}-u^{2}-c)^{2}}$
$\displaystyle-\delta_{2}^{-1}\sum_{j\in J}f_{j}-2\delta_{2}\sum_{j\in
J}f_{j}\Big{(}\frac{h_{11j}^{2}}{h_{11}^{2}}+\frac{(\nu^{n+1})_{j}^{2}}{(\nu^{n+1}-a)^{2}}\Big{)}.$

Taking (3.22)–(3.24) into (3.17), we obtain

(3.25)
$\displaystyle\Big{(}\kappa_{1}-\frac{1+(\nu^{n+1})^{2}}{\nu^{n+1}-a}-\frac{\beta\nu^{n+1}}{u}-\frac{2b(X\cdot\nu)u}{v^{2}-u^{2}-c}\Big{)}\psi-C$
$\displaystyle+\Big{(}\frac{\beta}{u}+\frac{a-2-4a^{-1}}{\nu^{n+1}-a}-1-\delta_{2}^{-1}\Big{)}\sum
f_{i}+\frac{3a}{4(\nu^{n+1}-a)}\sum f_{i}\kappa_{i}^{2}$
$\displaystyle-\Big{(}b+b^{2}+2(1+\delta_{2}^{-1})b^{2}\Big{)}\sum_{j\in
I}f_{j}\frac{(v^{2}-u^{2})_{j}^{2}}{(v^{2}-u^{2}-c)^{2}}$
$\displaystyle-(2b^{-1}+3\delta_{2})\frac{F^{ii}(\nu^{n+1})_{i}^{2}}{(\nu^{n+1}-a)^{2}}-\frac{4\beta^{2}}{b}\sum_{j\in
J}f_{j}\frac{u_{j}^{2}}{u^{4}}-2(1+\delta_{2}^{-1})\beta^{2}\sum_{j\in
I}f_{j}\frac{u_{j}^{2}}{u^{4}}$
$\displaystyle-\frac{F^{ij,rs}h_{ij1}h_{rs1}}{\kappa_{1}}-(1+4b^{-1}+2\delta_{2})\sum_{j\in
J}\frac{f_{j}h_{11j}^{2}}{h_{11}^{2}}\leq 0.$

Applying (3.21) and choosing $\delta_{2}=\frac{a^{2}}{16}$, (3.25) becomes

(3.26)
$\displaystyle\Big{(}\kappa_{1}-\frac{1+(\nu^{n+1})^{2}}{\nu^{n+1}-a}-\frac{\beta\nu^{n+1}}{u}-\frac{2b(X\cdot\nu)u}{v^{2}-u^{2}-c}\Big{)}\psi-C$
$\displaystyle+\Big{(}\frac{\beta}{u}+\frac{\frac{5a}{8}-2-4a^{-1}}{\nu^{n+1}-a}-1-\frac{16}{a^{2}}-\frac{4}{b(\nu^{n+1}-a)^{2}}-\frac{4\beta^{2}}{bu^{2}}\Big{)}\sum
f_{i}$
$\displaystyle+\Big{(}\frac{3a}{8(\nu^{n+1}-a)}-\frac{4}{b(\nu^{n+1}-a)^{2}}\Big{)}\sum
f_{i}\kappa_{i}^{2}-\frac{C\big{(}b+(3+32a^{-2})b^{2}\big{)}}{\theta^{2}(v^{2}-u^{2}-c)^{2}}f_{1}$
$\displaystyle-\frac{2(1+16a^{-2})\beta^{2}}{\theta^{2}u^{2}}f_{1}-\frac{F^{ij,rs}h_{ij1}h_{rs1}}{\kappa_{1}}-\Big{(}1+\frac{4}{b}+\frac{a^{2}}{8}\Big{)}\sum_{j\in
J}\frac{f_{j}h_{11j}^{2}}{h_{11}^{2}}\leq 0.$

By an inequality of Andrews [2] and Gerhardt [4], we have

$\displaystyle-\frac{F^{ij,rs}h_{ij1}h_{rs1}}{\kappa_{1}}\geq\frac{2}{\kappa_{1}}\sum_{j\in
J}\frac{f_{j}-f_{1}}{\kappa_{1}-\kappa_{j}}h_{11j}^{2}\geq
2(1-\theta)\sum_{j\in J}\frac{f_{j}h_{11j}^{2}}{h_{11}^{2}}.$

Choosing $\theta=\frac{1}{4}$, we have

$-\frac{F^{ij,rs}h_{ij1}h_{rs1}}{\kappa_{1}}-\Big{(}1+\frac{4}{b}+\frac{a^{2}}{8}\Big{)}\sum_{j\in
J}\frac{f_{j}h_{11j}^{2}}{h_{11}^{2}}\\\
\geq\Big{(}\frac{1}{2}-\frac{4}{b}-\frac{1}{8}\Big{)}\sum_{j\in
J}\frac{f_{j}h_{11j}^{2}}{h_{11}^{2}}.$

Also, choosing $\beta$ sufficiently large, (3.26) reduces to

$\displaystyle\Big{(}\kappa_{1}-\frac{1+(\nu^{n+1})^{2}}{\nu^{n+1}-a}-\frac{\beta\nu^{n+1}}{u}-\frac{2b(X\cdot\nu)u}{v^{2}-u^{2}-c}\Big{)}\psi-C$
$\displaystyle+\Big{(}\frac{\beta}{2u}-\frac{4}{b(\nu^{n+1}-a)^{2}}-\frac{4\beta^{2}}{bu^{2}}\Big{)}\sum
f_{i}+\Big{(}\frac{3a}{8(\nu^{n+1}-a)}-\frac{4}{b(\nu^{n+1}-a)^{2}}\Big{)}\sum
f_{i}\kappa_{i}^{2}$
$\displaystyle-\frac{C\big{(}b+(3+32a^{-2})b^{2}\big{)}}{(v^{2}-u^{2}-c)^{2}}f_{1}-\frac{C(1+16a^{-2})\beta^{2}}{u^{2}}f_{1}+\Big{(}\frac{3}{8}-\frac{4}{b}\Big{)}\sum_{j\in
J}\frac{f_{j}h_{11j}^{2}}{h_{11}^{2}}\leq 0.$

Then choosing $b$ sufficiently large, we have

$\displaystyle\Big{(}\kappa_{1}-\frac{1+(\nu^{n+1})^{2}}{\nu^{n+1}-a}-\frac{\beta\nu^{n+1}}{u}-\frac{2b(X\cdot\nu)u}{v^{2}-u^{2}-c}\Big{)}\psi-C$
$\displaystyle+\frac{3a}{16(\nu^{n+1}-a)}f_{1}\kappa_{1}^{2}-\frac{C\big{(}b+(3+32a^{-2})b^{2}\big{)}}{(v^{2}-u^{2}-c)^{2}}f_{1}-\frac{C(1+16a^{-2})\beta^{2}}{u^{2}}f_{1}\leq
0.$

We thus obtain an upper bound for $(v^{2}-u^{2}-c)\kappa_{1}$.

Now we have finished the proof of (3.7). In view of (3.5) and (3.6), we can
deduce that

(3.27)
$\max\limits_{\overline{\Omega_{\epsilon_{0}}}}\big{|}\kappa_{i}[u^{\epsilon}]\big{|}\,\leq\,C,\quad\forall\,\,i=1,\ldots,n,\quad\forall\,\,0<\epsilon<\frac{\delta_{\epsilon_{0}}}{2}.$

###  3.3. Uniqueness

Under assumption (1.7), we can prove the following uniqueness result.

######  Theorem 3.3.

Under assumption (1.7), the admissible solution $u\in C^{2}(\Omega)\cap
C^{0}(\overline{\Omega})$ to asymptotic Plateau problem (1.1) is unique.

###### Proof.

Let $u$ and $v$ be respectively an admissible solution and subsolution to
(1.1) in $\Omega$. We claim that $u\geq v$ in $\Omega$.

Suppose not. Then $v-u$ achieves a positive maximum at $x_{0}\in\Omega$.
Consequently, we have

$v(x_{0})>u(x_{0}),\quad Dv(x_{0})=Du(x_{0}),\quad D^{2}v(x_{0})\leq
D^{2}u(x_{0}).$

Now we consider the deformation $u[s]=sv+(1-s)u$ for $s\in[0,1]$. By direct
calculation, we can verify that $u[s]$ is admissible near $x_{0}$ for any
$s\in[0,1]$. This is because at $x_{0}$,

$\displaystyle\delta_{ij}+u[s]{\gamma}^{ik}(u[s])_{kl}\gamma^{lj}\geq\delta_{ij}+u[s]{\gamma}^{ik}v_{kl}\gamma^{lj}$
$\displaystyle=$
$\displaystyle(1-s)\Big{(}1-\frac{u}{v}\Big{)}\delta_{ij}+\frac{u[s]}{v}\Big{(}\delta_{ij}+v\gamma^{ik}v_{kl}\gamma^{lj}\Big{)}.$

Note that equation (1.1) can be rewritten as

$f(\kappa[u])=F(A[u])=G(D^{2}u,Du,u)=G[u]=\psi(x,u).$

Denote

$G^{ij}=\frac{\partial G}{\partial u_{ij}},\quad G^{i}=\frac{\partial
G}{\partial u_{i}},\quad G_{u}=\frac{\partial G}{\partial u}.$

Now we can define a differentiable function for $s\in[0,1]$,

$a(s)=G(D^{2}u[s],Du[s],u[s])(x_{0})-\psi(x_{0},u[s](x_{0})).$

Since $a(0)=0$ and $a(1)\geq 0$, there exists $s_{0}\in[0,1]$ such that
$a(s_{0})=0$ and $a^{\prime}(s_{0})\geq 0$, or equivalently,

$G\big{[}u[s_{0}]\big{]}(x_{0})=\psi(x_{0},u[s_{0}](x_{0})),$

and

(3.28) $\displaystyle
G^{ij}\big{[}u[s_{0}]\big{]}(x_{0})D_{ij}(v-u)(x_{0})+G^{i}\big{[}u[s_{0}]\big{]}(x_{0})D_{i}(v-u)(x_{0})$
$\displaystyle+\big{(}G_{u}\big{[}u[s_{0}]\big{]}(x_{0})-\psi_{u}\big{(}x_{0},u[s_{0}](x_{0})\big{)}\big{)}(v-u)(x_{0})\geq
0.$

Also note that

$G^{ij}=\frac{u}{w}F^{kl}\gamma^{ki}\gamma^{jl},\quad
G_{u}=\frac{1}{u}\big{(}\psi-\frac{1}{w}\sum f_{i}\big{)}.$

Thus,

$G^{ij}\big{[}u[s_{0}]\big{]}(x_{0})D_{ij}(v-u)(x_{0})\leq 0$

and

$G_{u}\big{[}u[s_{0}]\big{]}(x_{0})-\psi_{u}(x_{0},u[s_{0}](x_{0}))<0$

by assumption (1.7). But then (3.28) can not hold. Thus, we proved the claim.
Consequently, we obtain the uniqueness. ∎

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