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\begin{document}
\title[A generalized Minkowski problem]{A 
generalized Minkowski problem\linebreak[1]  with Dirichlet boundary condition}
\author{Oliver C. Schn\"urer}
\address{Max Planck Institute for Mathematics in the Sciences,
Inselstrasse 22--26, D-04103 Leipzig, Germany}
\curraddr{Department of Mathematics, Harvard University, 
One Oxford Street, Cambridge, MA 02138, USA}
\email{Oliver.Schnuerer@mis.mpg.de}
\email{schnuere@math.harvard.edu}
\subjclass[2000]{Primary 35J65; Secondary 53C42}
\date{November 8, 2000}
\keywords{Minkowski problem, Dirichlet problem, 
prescribed curvature, convex hypersurfaces}
\begin{abstract}
We prove the existence of hypersurfaces with prescribed
boundary whose Weingarten curvature equals a given function
that depends on the normal of the hypersurface.
\end{abstract}
\maketitle

\section{Introduction}
Let $\tilde M\in\myrone$ be a strictly convex smooth hypersurface
and $0<f\in C^{2,\alpha}(S^n)$. We denote by $\tilde M_+$
an open subset of $\tilde M$ with smooth boundary
$\partial\tilde M_+$ (with respect to $\tilde M$) and
$\tilde M_+\subsetneq\tilde M$. Let $F$ be a curvature function,
as, for example $S_n$, the Gau{\ss} curvature, or $S_k\cdot S_n$,
$1\le k\le n$, where $S_k$ denotes the $k$-th elementary
symmetric polynomial
\begin{equation}
S_k(\kappa_i)=\sum\limits_{1\le i_1<\ldots<i_k\le n}
\kappa_{i_1}\cdot\ldots\cdot\kappa_{i_k}.
\end{equation}
A description of the curvature functions considered in this
paper is given in Section \ref{curvaturefunctionsection}.
We denote the principal curvatures of a hypersurface $M$
by $\kappa_i(M)$, $1\le i\le n$, and the outer unit normal
of $\tilde M$ by $\nu_{\widetilde M}$. Under these assumptions we prove
the following theorem. 
\begin{theorem}
If $F(\kappa_i(\tilde M))\ge f(\nu_{\tilde M})$ in $\tilde M_+$,
then there exists a $C^{4,\alpha}$-hypersurface $M$ with
\begin{equation}
F(\kappa_i(M))=f(\nu)
\end{equation}
and $\partial M=\partial\tilde M_+$, where $\nu$ denotes an
appropriately chosen unit normal of $M$.
\end{theorem}

More precisely, we look for a strictly convex hypersurface
contained in the convex body determined by $\tilde M$,
whose unit normal points inside the bounded component
of $\myrone\setminus\ol{(\tilde M_+\cup M)}$ and
$M\cup(\tilde M\setminus\tilde M_+)$ is the boundary of a
strictly convex body.

The paper is organized as follows. In Section
\ref{curvaturefunctionsection} we introduce the curvature
functions we are interested in, and in Section \ref{Notations} we
state our notation for differential geometry.
Simple a priori estimates are proved in Section 
\ref{Preliminary a priori estimates}, and more complicated
ones are proved in Section \ref{A priori estimates}. Finally,
we sketch in Section \ref{Existence} how these a priori estimates
can be used for the existence proof.

The classical Minkowski problem has been solved for smooth functions
in any dimension in \cite{cheng yau}. A generalization for other 
curvature functions has been obtained recently in 
\cite{guanguanpre2000}. There are many papers that consider 
a Dirichlet problem for equations of prescribed curvature.
The most useful ones for our estimates were \cite{tr} and
\cite{bg}. We remark that our allowing of the dependence of $f$ on $\nu$ 
without imposing structure conditions on $f$ besides positivity
and smoothness seems to be new. 

We wish to thank J\"urgen Jost and the Max Planck Institute
for Mathematics in the Sciences, Leipzig, where this research
was carried out, for their hospitality. We would also like to
thank Claus Gerhardt, Heidelberg, who brought the
Minkowski problem to our attention.     

\section{Curvature Functions}
\label{curvaturefunctionsection}
In this paper, we consider symmetric curvature functions defined
in the positive cone $\Gamma_+\subset\myR^n$ which are of the class
$C^{2,\alpha}(\Gamma_+)\cap C^0\left(\ol{\Gamma_+}\right)$,
positive homogeneous and satisfy
\begin{align}
F_i=\fracp{F}{\kappa_i}&>0 \mbox{ in } \Gamma_+,\\
\left.F\right|_{\partial\Gamma_+}&=0.\nonumber
\end{align}
For a positive definite symmetric matrix
$(h_{ij})\in\sym(n,\myR)$ with eigenvalues $\kappa_i$, $1\le i\le n$,
we define
\begin{equation}
F(h_{ij}):=F(\kappa_i),
\end{equation}
where the ambiguous notation should not cause any difficulties.
Given $F\in 
\linebreak  
C^{2,\alpha}(\sym)\cap C^0\left(\ol{\sym}\right)$, 
we can define
\begin{equation}
F^{ij}=\fracp{F}{h_{ij}},\quad
F^{ij,kl}=\frac{\partial^2F}{\partial h_{ij}\partial h_{kl}},
\end{equation}
so that, for an appropriate interpretation of the right-hand side,
\begin{equation}
F^{ij}\xi_i\xi_j=\fracp{F}{\kappa_i}|\xi_i|^2\quad
\forall\,\xi\in\myR^n
\end{equation}
and $F^{ij}$ is diagonal if $h_{ij}$ is diagonal.

We define the elementary symmetric polynomials by
\begin{equation}
S_k(\kappa):=\sum\limits_{1\le i_1<\ldots<i_k\le n}
\kappa_{i_1}\cdot\ldots\cdot\kappa_{i_k},\quad 1\le k\le n,
\end{equation}
and furthermore,
\begin{align}
\tilde S_k(\kappa)&:=
  \frac{1}{S_k\left(\frac{1}{\kappa_i}\right)}
  =\frac{S_n(\kappa)}{S_{n-k}(\kappa)},\\
\sigma_k&:=(S_k)^{1/k},\quad
  \tilde\sigma_k:=\left(\tilde S_k\right)^{1/k},\nonumber\\
\gamma_k&:=
  \left(\sum\limits_{|\alpha|=k}\kappa^\alpha\right)^{1/k},
  \quad\tilde\gamma_k(\kappa):=
  \frac{1}{\gamma_k\left(\frac{1}{\kappa_i}\right)},\nonumber
\end{align}
where $\alpha$ is a multi-index.

In this paper we will consider curvature functions $F$ of the
form
\begin{equation}
F=\sigma^\alpha\cdot\gamma^{\alpha'},\quad\mbox{where }
\sigma^\alpha\equiv\prod\limits_{k=1}^n\sigma_k^{\alpha_k},
\end{equation}
and we assume $\alpha_n>0$ and $\alpha,\,\alpha'\ge 0$ in the
sense of multi-indices. These functions obviously fulfill the
conditions for curvature functions from the beginning of this
section, and, furthermore,
\begin{equation}
\tilde F(\kappa_i):=\frac{1}{F\left(\frac{1}{\kappa_i}\right)}
  =\tilde\sigma^\alpha\cdot\tilde\gamma^{\alpha'}.
\end{equation}

Defining an abstract class of curvature functions for which the
existence proof can be carried out seems to require many
restrictions; so we mention only these examples. The fact that
$F$ contains a positive power of the Gau{\ss} curvature
is restrictive, but we have to ensure
$\left.F\right|_{\partial\Gamma_+}=0$ to preserve the convexity
of the hypersurfaces appearing in our existence proof as well as
a kind of concavity of $F$ to deduce $C^{2,\alpha}$ estimates
from the $C^2$ a priori estimates.

\begin{lemma}\label{lemthreeequivalent}
For $F$ as above, the function 
\begin{equation}
\hat{\tilde F}=\log\tilde F=-\log F\left(\frac{1}{\kappa_i}\right)
\end{equation}
is concave and the following properties are equivalent provided
$0<c\le F\le\frac{1}{c}<\infty$:
\begin{eqnarray}
\kappa_1&\to&0,\\
\kappa_n&\to&+\infty,\nonumber\\
\trF=F^{ij}\delta_{ij}&\to&+\infty,\nonumber
\end{eqnarray}
when $0<\kappa_1\le\ldots\le\kappa_n$.
\end{lemma}
\begin{proof}
The concavity follows from the properties of curvature
functions collected in \cite{cgjdg1996}.

$\kappa_1\to 0$ and $\left.F\right|_{\partial\Gamma_+}=0$
force $\kappa_n\to\infty$. $\kappa_1\ge c>0$ and
$\kappa_n\to\infty$ contradict $F\le\frac{1}{c}$. Here
and below we use $c$ to denote a positive constant that
may change its value if necessary. $\trF\to\infty$ is
impossible for $\kappa$ in a compact subset of $\Gamma_+$,
and so $\kappa_1\to0$ or $\kappa_n\to\infty$.

Let $F=\sigma^\alpha\cdot\gamma^{\alpha'}$ as above. Then 
\begin{equation}
\trF\ge\frac{\alpha_n}{n}\cdot\sigma^\alpha\cdot
\frac{1}{\kappa_1}\cdot\gamma^{\alpha'}
\ge\frac{\alpha_n}{n}\cdot c\cdot\frac{1}{\kappa_1}.
\end{equation}
Therefore, $\kappa_1\to0$ implies $\trF\to+\infty$.\end{proof}

We may assume that $F$ is positive homogeneous of degree one.

When we consider a hypersurface $M$ with induced metric
$g_{ij}$, we compute the eigenvalues of $h_{ij}$ with
respect to $g_{ij}$, or equivalently we compute the
eigenvalues of $h_{ik}g^{kj}\equiv h^j_i$,
$(g^{ij})=(g_{ij})^{-1}$. Then the chain rule yields
\cite{ocsmz200x}
\begin{equation}
\fracp{F}{g_{ij}}=-F^{ik}h^j_k.
\end{equation}
 
\section{Notation}
\label{Notations}
In Euclidean coordinate systems the induced metric $g_{ij}$
and the inverse $g^{ij}$ of the induced metric
of $\graph u$ are given by
\begin{eqnarray}
g_{ij}&=&\delta_{ij}+u_iu_j,\\
g^{ij}&=&\delta^{ij}-\frac{u^iu^j}{v^2},\nonumber\\
u^i&=&u_j\delta^{ij},\nonumber\\
v&=&\sqrt{1+|Du|^2}\ =\ \sqrt{1+u_iu^i},\nonumber
\end{eqnarray}
where Latin indices range from $1$ to $n$ and refer to quantities
of $\graph u$, except for $r$, $s$ and $t$, which only range to $n-1$
and refer to an $(n-1)$-dimensional boundary. The Einstein
summation convention is always used. Greek indices range from $0$
to $n$ and refer to $\myrone$.

Let $x^\alpha$ be a Euclidean coordinate system for $\myrone$, $\nu^\alpha$
the ``upwards pointing'' unit normal of $\graph u$,
$\Gamma^\cdot_{\cdot\cdot}$ the induced Christoffel symbols and
$h_{ij}$ the second fundamental form. We will choose coordinates
such that $h_{ij}$ is positive definite for a strictly convex
hypersurface. These quantities are
related by the Gau{\ss} formula ($x^\alpha_{;ij}$ indicate
covariant derivatives, while $u_{ij}$ and $x^\alpha_{,ij}$ denote
partial derivatives):
\begin{eqnarray}
x^\alpha_{;ij}&=&-h_{ij}\nu^\alpha,\\
\nu^\alpha&=&\frac{1}{v}(-u^i,1),\nonumber\\
x^\alpha_{;ij}&=&x^\alpha_{,ij}-\Gamma^k_{ij}x^\alpha_k,\nonumber\\
\Gamma^k_{ij}&=&\frac{1}{v^2}u^ku_{ij},\nonumber\\
h_{ij}&=&-\left(u_{ij}-\Gamma^k_{ij}u_k\right)\cdot v
  \ =\ -\frac{u_{ij}}{v}.\nonumber
\end{eqnarray}

As $M$ is strictly convex, $M\ni z\mapsto\nu(z)$ is a
diffeomorphism from $M$ to $\nu(M)\subset S^n$.
For $x\in\nu(M)$ we define the supporting function $u$
(the ambiguous use of $u$ should not cause any difficulties)
by
\begin{equation}
u(x)=x^\alpha\delta_{\alpha\beta}\left(\nu^{-1}\right)^\beta(x)
=\langle x,\nu^{-1}(x)\rangle,
\end{equation}
i.e., the scalar product is the usual scalar product in $\myrone$.
We remark that the original hypersurface can be recovered from
the support function \cite{cheng yau}. The eigenvalues of
\begin{equation}
(u_{;ij}+u\sigma_{ij})(x),\quad x\in S^n,
\end{equation}
with respect to the standard metric $\sigma_{ij}$ of $S^n$ are
the inverses of the principal curvatures of $M$ at
$\nu^{-1}(x)$, where $u_{;ij}$ denotes covariant derivatives
with respect to the metric $\sigma_{ij}$.
For covariant derivatives on $S^n$ we have
\begin{equation}
u_{;ijkl}=u_{;klij}+2u_{;ij}\sigma_{kl}
  -2u_{;kl}\sigma_{ij}+u_{;kj}\sigma_{il}
  -u_{;il}\sigma_{kj}.
\end{equation}


\section{Preliminary a Priori Estimates}
\label{Preliminary a priori estimates}
We assume that $M$ is a prospective solution and prove a priori
estimates for this solution. Later on, we will deform our problem
and use these estimates. Therefore, we will have to ensure
especially that $M$ lies inside the convex body determined by
$\tilde M$. This can be achieved if we approximate $f$ so 
that $\tilde M$ is a strict supersolution,
\begin{equation}
F(\kappa_i(\tilde M))>f(\nu_{\tilde M}).
\end{equation}
In view of our a priori estimates, we may afterwards choose a
subsequence of solutions of the modified problems that
converge to a solution of the original problem.

\subsection{$C^0$- and $C^1$-estimates}
\label{$C^0$- and $C^1$-estimates}
It is possible to represent $M$ as a graph over an appropriately
chosen part of a small sphere with center in
$\tilde M\setminus\ol{\tilde M_+}$, 
$\graph \left.u\right|_\Omega=M$,
such that $|u|_1$ is bounded. The $C^0$ bounds follow from the
geometric setting and the convexity of $M$; the $C^1$ bounds
follow as the angle between each half-line starting from the
center of the small sphere and $M$ in an intersection point
is a priori bounded from below. In our setting, this follows
by simple geometric reasoning. A proof for a much more general
situation can be found in \cite{tnjram1998}.

\subsection{A special coordinate system}
\label{A special coordinate system}
In view of the uniform $C^1$-estimates, we may rotate our
original Euclidean coordinate system appropriately so that
we have the following situation:
An arbitrary but fixed point $x_0$ of $\partial\tilde M_+$
is the origin of our coordinate system. Furthermore,
we have a function $\omega:\hat{\hat B}_r\to\myR$,
$\hat{\hat B}_r:=\{x''\in\myR^{n-1}: |x''|<r\}$, $\omega(0)=0$,
$D\omega(0)=0$, such that $M\cap\left(\hat B_r\times\myR\right)=
\graph \left.u\right|_\Omega$, where $\hat B_r:=
\{x'\in\myR^n:|x'|<r\}$, $\Omega:=\hat B_r\cap\{(x',x^n):
x^n>\omega(x')\}$ and $u$ is an appropriate function
whose graph locally coincides with $M$. $r>0$ is uniformly
bounded from below by a positive constant which is, in particular,
independent of $x_0$ and the second fundamental
form of $M$. $\omega$ and its derivatives are a priori bounded
as well as $|u|_1$. The function $u$ is as smooth as $M$, but
its derivatives of order greater than one are
not yet a priori bounded. In the same way in which $M$ is locally
represented as $\graph u$, we may assume that $\tilde M=
\graph \left.\tilde u\right|_\Omega$, but the derivatives of
$\tilde u$ are a priori bounded. Furthermore, we may
assume $u(0)=\tilde u(0)=0$ and $u_r(0)=\tilde u_r(0)=0$,
$1\le r\le n-1$.

\subsection{Tangential $C^2$-estimates at the boundary}
We choose a coordinate system as described above.
On $\partial\Omega$, $u$ and $\tilde u$ coincide;
so we may differentiate
\begin{equation}
(\tilde u-u)(x'',\omega(x''))=0
\end{equation}
twice and obtain (because $D\omega(0)=0$ for $1\le r,s\le n-1$
at the origin)
\begin{equation}
u_{rs}=\tilde u_{rs}+(\tilde u-u)_n\omega_{rs}.
\end{equation}
All terms on the right-hand side are bounded. So $|u_{rs}|$,
i.e., the tangential derivatives of $u$, are a priori bounded.

\section{A Priori Estimates}
\label{A priori estimates}
Compare this proof to the respective ones in \cite{cnsi},
\cite{cnsiii}, \cite{bgjs}, \cite{bg}, \cite{tr} and
\cite{ocsmz200x}.

\subsection{Mixed $C^2$-estimates at the boundary}
\label{Mixed estimates at the boundary}
We consider the situation for a fixed point of $\partial M$
in a coordinate system as described in Section
\ref{A special coordinate system}.

We differentiate the equation
\begin{equation}
F(h_{ij},g_{ij})=f(\nu^\alpha)\equiv f(u_i)
\end{equation}
with respect to $x^k$ and use
\begin{equation}
h_{ij,k}=-\frac{1}{v}u_{ijk}+\frac{1}{v^3}u^lu_{lk}u_{ij};
\end{equation}
thus,
\begin{equation}
f_{p_i}u_{ik}=-\frac{1}{v}F^{ij}u_{ijk}
  -\frac{1}{v^2}Fu^lu_{lk}
  -F^{il}h^j_l(u_{ik}u_j+u_iu_{jk}).
\end{equation}
Therefore we define the linear operator $L$ by
\begin{equation}
Lw:=\frac{1}{v}F^{ij}w_{ij}+\frac{1}{v^2}Fu^lw_l
  +F^{il}h^j_l(u_jw_i+u_iw_j)+f_{p_i}w_i,
\end{equation}
and for $t<n$ we define 
\begin{equation}
T:=\fracp{}{x^t}+B_{tr}x^r\fracp{}{x^n}-B^r_tx_n\fracp{}{x^r},
\end{equation}
where
\begin{equation}
\omega(x')=\tfrac{1}{2}B_{rs}x^rx^s+O\left(|x'|^3\right),\quad
  x'=(x^1,\ldots,x^{n-1}),
\end{equation}
$B^r_t\equiv B_{ts}\delta^{rs}$, $x_n=x^i\delta_{in}$ and
$r$, $s$ and $t$ run from $1$ to $n-1$ as usual.

From the definition of $L$ and homogeneity, it is easy to
see that
\begin{equation}
|LT(u-\tilde u)|\le c\cdot\trFe,
\end{equation}
where, here and in the following, $c$ is bounded from above
by estimated quantities.

Furthermore, in view of the $C^1$-estimates we have 
\begin{equation}
|T(u-\tilde u)|\le c\quad\mbox{in }\ol{\Omega},
\end{equation}
and, due to the definition of $T$,
\begin{equation}
|T(u-\tilde u)|\le c\cdot|x|^2\quad\mbox{on }\partial\Omega.
\end{equation}

We consider the function
\begin{equation}
\vartheta:=(\tilde u-u)+\alpha d-\mu d^2
\end{equation}
in a domain $\Omega_\delta:=\Omega\cap B_\delta(0)$, where $d$
is the distance function from $\partial\Omega$ in $\myR^n$, and
show that it satisfies a nice differential equation for $\alpha$,
$\delta>0$ small and $\mu$ large.

As $\graph\tilde u$ is strictly convex, we fix $\varepsilon>0$
such that $-\tilde u_{ij}\ge\varepsilon\delta_{ij}$ in the
matrix sense. We compute
\begin{equation}
L\vartheta\le-\frac{\varepsilon}{v}\trF-2\mu\frac{1}{v}F^{ij}d_id_j
  +c+c\cdot(\alpha+\delta\mu)\cdot\trFe.
\end{equation}

We use 
$-F^{rs}d_rd_s\le0$, $1\le r,s\le n-1$, as well as
$|F^{nr}|\le\frac{1}{2}\trF$ and $d_i\approx\delta_{in}$; so
\begin{equation}
L\vartheta\le-\frac{\varepsilon}{v}\trF-\mu\frac{1}{v}F^{nn}
  +c+c\cdot(\alpha+\delta\mu)\cdot\trFe.
\end{equation}
From Lemma \ref{lemthreeequivalent}, we infer that for sufficiently
large fixed $\mu$ the constant $c$ can be absorbed by 
the first two terms on the right-hand side. 
For small $\alpha,\,\delta>0$
we obtain
\begin{equation}
L\vartheta\le-\frac{1}{3}\frac{\varepsilon}{v}\trF.
\end{equation}
Furthermore, when $\delta>0$ is small,
\begin{equation}
\vartheta\ge0\quad\mbox{on }\partial\Omega_\delta.
\end{equation}

We consider the function
\begin{equation}
\Theta:=A\vartheta+B|x|^2\pm T(u-\tilde u).
\end{equation}
We fix $B\gg1$, get $\Theta\ge0$ on $\partial\Omega_\delta$, and
deduce from Lemma \ref{lemthreeequivalent} that  $\trF\ge\frac{1}{c}>0$;
so $L\Theta\le0$ for $A\gg B$. The maximum principle yields
$\Theta\ge0$ in $\Omega_\delta$. As $\Theta(0)=0$, we have
$\Theta_n(0)\ge0$, which in turn immediately gives 
$|u_{tn}(0)|\le c$, $1\le t\le n-1$.

\subsection{Normal $C^2$-estimates at the boundary}
We may assume that the infimum of the invariantly defined
function
\begin{equation}
\partial\Omega\ni x\mapsto
\inf\limits_{0\not=\zeta\in T_x\partial\Omega}
\frac{h_{ij}\zeta^i\zeta^j(x)}{g_{ij}\zeta^i\zeta^j(x)}
\end{equation}
equals $h_{11}(x_0)/g_{11}(x_0)$, and fix a coordinate system
around $x_0$ as in Section \ref{A special coordinate system}.
We choose smooth vector fields $\xi_i$, $1\le i\le n$, around
$x_0$ such that $\xi_n$ equals the inner unit normal to
$\partial\Omega$, $\xi_1(x_0)=e_1(x_0)$, and the vectors
$\xi_i$ are pointwise orthonormal with respect to the
Euclidean metric.

From $\tilde u-u=0$ on $\partial\Omega$, we deduce along
$\partial\Omega$ the well-known relation \cite{tr}
\begin{eqnarray}
\label{Trudinger relation}
0&=&\xi^i_r\xi^j_s(\tilde u-u)_{ij}
  +\xi^i_n(\tilde u-u)_i
  \left(-\xi^l_r\xi^j_s\delta_{jk}
  \left(\xi^k_n\right)_l\right)\\
&\equiv&\nabla_{rs}(\tilde u-u)+(\tilde u-u)_\nu C_{rs}.
  \nonumber
\end{eqnarray}
We wish to estimate $-\nabla_{11}u(x_0)$ from below by a
positive constant. If $-\nabla_{11}u(x_0)\ge
-\frac{1}{2}\nabla_{11}\tilde u(x_0)$, we have such an
estimate; otherwise we deduce from (\ref{Trudinger relation}) 
that $C_{11}$ is locally bounded from
below by a positive constant. As $|\nabla_{11}u|$ is
already bounded on $\partial\Omega$, this means that 
\begin{equation}
\partial\Omega\ni x\mapsto-\nabla_{11}u+a|x|^2
\end{equation}
attains its infimum in $x_1$
near $x_0=0$ for $a\gg1$, and thus
\begin{equation}
c(x_1)\equiv-\nabla_{11}u(x_1)+a|x_1|^2\le
  -\nabla_{11}u(x)+a|x|^2,\quad x\in\partial\Omega,
\end{equation}
or, in view of (\ref{Trudinger relation}), for 
$x\in\partial\Omega$,
\begin{align}
u_\nu(x)&\ge C_{11}^{-1}(x)\cdot
  (\nabla_{11}\tilde u(x)+c(x_1)-a|x|^2)+\tilde u_\nu(x)\\
&\equiv\gamma\in C^2,\quad |\gamma|_2\le c. \nonumber
\end{align}
For $\Theta:=A\vartheta+B|x-x_1|^2-\gamma(x)+u_\nu(x)$,
when $\gamma$ is extended appropriately, we deduce
as in Section \ref{Mixed estimates at the boundary}
that $-\nabla_{nn}u(x_1)$ is bounded from above.
Lemma \ref{lemthreeequivalent} implies a positive lower
bound for $-\nabla_{11}u(x_1)$, thus also for
$-\nabla_{11}u(x_0)$, $h_{11}(x_0)/g_{11}(x_0)$ and
$h_{\zeta\zeta}/g_{\zeta\zeta}$, $\zeta\in T\partial\Omega$.
In view of Lemma 1.2 in \cite{cnsiii} (or Young's inequality)
and Lemma \ref{lemthreeequivalent}, we deduce a bound for the
second derivatives of $u$ on $\partial\Omega$, and
$0<\frac{1}{c}\le\kappa_i\le c$ for all eigenvalues $\kappa_i$
of $h_{ik}g^{kj}$.

\subsection{Interior $C^2$-estimates}
We proceed as in \cite{guanguanpre2000} and transfer the 
situation to $S^n$ via the Gau{\ss} map
$M\ni x\mapsto\nu(x)\in S^n$.

For the estimates here, indices denote covariant derivatives
with respect to $\sigma_{ij}$, $u$ is the support function
of $M$, and we remark that
\begin{eqnarray}
F(h_{ik}g^{kj})&=&\left(\tilde F((u_{ik}+u\sigma_{ik})
  \sigma^{kj})\right)^{-1},\\
\tilde F((u_{ik}+u\sigma_{ik})(x)&=&(f(x))^{-1},\quad
  x\in\nu(M)\subset S^n.\nonumber
\end{eqnarray}

On $\partial M$ the eigenvalues of the second fundamental form
are a priori bounded from below and from above. Assume that
\begin{equation}
\frac{1}{\kappa_1(M)}+\ldots+\frac{1}{\kappa_n(M)}=
\sigma^{ij}(u_{ij}+u\sigma_{ij})\equiv\sigma^{ij}w_{ij}
\end{equation}
attains its maximum at an interior point 
$\nu^{-1}(x_0)$ of $M$. So we have there
\begin{eqnarray}
0&\ge&(\sigma^{ij}w_{ij})_{kl}\\
&=&\sigma^{ij}(u_{ijkl}+u_{kl}\sigma_{ij})\nonumber\\
&=&\sigma^{ij}(u_{klij}+2u_{ij}\sigma_{kl}-u_{kl}\sigma_{ij}
  +u_{kj}\sigma_{il}-u_{il}\sigma_{kj})\nonumber\\
&=&\Delta(u_{kl})+2\Delta u\sigma_{kl}-nu_{kl}\nonumber\\
&=&\Delta(w_{kl})-nw_{kl}+\sigma^{ij}w_{ij}\sigma_{kl},\nonumber
\end{eqnarray}
where $\Delta(\cdot)=\sigma^{ij}(\cdot)_{ij}$. Define
\begin{equation}
{\hat{\tilde F}}^{kl}=\fracp{\hat{\tilde F}}{w_{kl}}(w_{ij}).
\end{equation}
As ${\hat{\tilde F}}^{ij}$ is positive,
\begin{align}
0&\ge {\hat{\tilde F}}^{kl}(\sigma^{ij}w_{ij})_{kl} \\
&={\hat{\tilde F}}^{kl}\Delta(w_{kl})-n{\hat{\tilde F}}^{kl}w_{kl}
  +\sigma^{ij}w_{ij}{\hat{\tilde F}}^{kl}\sigma_{kl}. \nonumber
\end{align}
$\tilde F$ is homogeneous of degree one, and  so
\begin{equation}
{\hat{\tilde F}}^{kl}w_{kl}=
\frac{1}{\tilde F}\fracp{\tilde F}{w_{kl}}w_{kl}=1.
\end{equation}
We differentiate $\hat{\tilde F}(w_{ik}\sigma^{kj})=-\log f(x)$
and use the concavity
\begin{equation}
\sigma^{ij}{\hat{\tilde F}}^{kl}w_{kl;ij}\ge-\Delta\log f(x),
\end{equation}
and so we obtain
\begin{equation}
0\ge-\Delta\log f-n+(\sigma^{ij}w_{ij})\cdot
{\hat{\tilde F}}^{kl}\sigma_{kl}.
\end{equation}
As ${\hat{\tilde F}}^{kl}\sigma_{kl}$ is bounded from below
by a positive constant (show that this quantity tends to
infinity when an eigenvalue of $w_{ij}$ approaches zero as
in the proof of Lemma \ref{lemthreeequivalent}) and $f$
is a given function, we conclude that $\sigma^{ij}w_{ij}$
is a priori bounded, i.e., $\kappa_i(M)$, $1\le i\le n$,
is bounded from below by a positive constant, and in view
of Lemma \ref{lemthreeequivalent} also from above. This
proves the a priori estimate for the eigenvalues of the
second fundamental form and for the second derivatives
of a function representing $M$ locally as a graph in
coordinate systems as in Section 
\ref{$C^0$- and $C^1$-estimates}
or \ref{A special coordinate system}.

\subsection{$C^{2,\alpha}$-estimates and further estimates}
As mentioned in \cite{cgjdg1996}, the functions $\sigma_k$ and
$\gamma_k$, $1\le k\le n$, are concave and monotone increasing.
Our a priori estimates now guarantee that 
\begin{equation}
\log F=\sum\limits_k\alpha_k\log\sigma_k
+\sum\limits_k\alpha'_k\log\gamma_k
\end{equation}
is uniformly elliptic and strictly concave.
So we can apply Krylov-Safanov theory,
Chapter 14.13 in \cite{TaylorPDE3}, and deduce $C^{2,\alpha}$
a priori estimates. Finally, we deduce $C^{4,\alpha}$ a priori
estimates by using Schauder theory.

\section{Existence}
\label{Existence}
To prove the existence of the hypersurface $M$ that  we are looking
for, we may proceed as in \cite{it} or \cite{tnjram1998}
and deform our problem into a local problem such that $M$
and $\tilde M_+$ are representable as graphs in a single
Euclidean coordinate system. During this step we represent
solutions of the deformed problem in a coordinate system
as in Section \ref{$C^0$- and $C^1$-estimates}.
If this deformation is carried
out appropriately, the degree mod $2$ implies that it suffices
to solve the local problem. The maximum principle yields that
solutions of the deformed problem fulfill analogous geometric
conditions as stated next to the main theorem. In view of the
uniqueness of a prospective solution of the local problem and
the a priori estimates, a standard continuity method yields
the existence for the local problem.
This in turn yields a solution of our original problem
via degree mod $2$ theory.

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\end{document}
