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\newtheorem{theorem}{Theorem}[section] 
\newtheorem{lemma}[theorem]{Lemma} 
\newtheorem{corollary}[theorem]{Corollary} 
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%    Absolute value notation 
\newcommand{\abs}[1]{\lvert#1\rvert} 
 
%%%%% Eigene Definitionen %%%%% 
\def\graph{{\rm graph\, }} 
\def\R{{\mathbb{R}}} 
\def\N{{\mathbb{N}}}
\def\Z{{\mathbb{Z}}}
\def\tr{{\rm tr\,}} 
\def\tru{{\rm tr\,}u^{ij}} 
\def\trF{{{\rm tr\,}\hat F^{ij}}}
\def\trup{\left(1+{\rm tr\,}u^{ij}\right)} 
\def\trupD{\left(1+\left|D^2u\right|+{\rm tr\,}u^{ij}\right)}
\def\dist{{{\rm dist\,}}}
\def\ol#1{\overline{#1}}
\def\fracp#1#2{\frac{\partial #1}{\partial #2}}
% Abkuerzung steht fuer n+2
\def\npz#1{{\left(1+|#1|^2\right)^{\frac{n+2}{2}}}} 
\def\mnpz#1{{\left(1+|#1|^2\right)^{-\frac{n+2}{2}}}} 
\def\dfrac#1#2{{\displaystyle\frac{\displaystyle #1}%
{\displaystyle #2}}}
\def\ol#1{{\overline{#1}}}
\def\osc{{\rm osc\,}}
\def\Kstar{{\left(\tilde K^\star\right)}}
\def\Ffv{{\frac{\hat F-\hat f}{v}}}
\def\Ffvkl{{\left(\frac{\hat F-\hat f}{v}\right)}}
\def\diam{{\rm diam\,}}

\begin{document}  

\title{Translating solutions to the second boundary value problem
  for curvature flows}
 
%    Information for first author 
\author{Oliver C.\ Schn\"urer} 
%    Address of record for the research reported here 
\address{Max-Planck-Institute for Mathematics in the Sciences,  
  Inselstr.\ 22-26, 04103 Leipzig, Germany}
\email{Oliver.Schnuerer@mis.mpg.de}
%    Current address 
\curraddr{Harvard University, Department of Mathematics,
  One Oxford Street, Cambridge, MA 02138, USA} 
\email{schnuere@math.harvard.edu}
%    \thanks will become a 1st page footnote. 
%\thanks{} 
 
%    Information for second author 
%\author{} 
%\address{Max-Planck-Institute for Mathematics in the Sciences, 
%  Inselstr.\ 22-26, 04103 Leipzig, Germany} 
%\email{@mis.mpg.de} 
%\thanks{} 
 
%    General info 
\renewcommand{\subjclassname}{%
  \textup{2000} Mathematics Subject Classification}
\subjclass{Primary 53C44; Secondary 35K20, 53C42} 
 
\date{May 2001 and January 2002.} 
 
%\dedicatory{} 
 
%\keywords{} 
 
\begin{abstract}
We consider the flow of strictly convex hypersurfaces 
driven by curvature functions subject to the second
boundary condition and show that they converge to 
translating solutions. We also discuss translating 
solutions for Hessian equations.
\end{abstract} 
 
\maketitle 
 
\markboth{Oliver C.\ Schn\"urer}
{Translating solutions to the second boundary value problem
  for curvature flows}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
We consider the parabolic initial value problem
describing the evolution of a hypersurface in $\R^{n+1}$
\begin{equation}\label{flow eqn}
\left\{\begin{array}{r@{~=~}l}
\dot X & -(\log F-\log f)\nu,\\[0.3ex]
\nu(M) & \nu(M_0),\\[0.3ex]
\left.M\right|_{t=0}& M_0,
\end{array}\right.
\end{equation}
where $X$ is the embedding vector of a smooth strictly convex 
hypersurface with boundary, $M=\graph \left.(-u)\right|_\Omega$,
$u:\ol{\Omega}\to\R$, 
$\dot X$ is its total time derivative and
$\Omega\subset\R^n$, $n\ge2$, 
is a smooth strictly convex domain. 
The fact that $M=\graph(-u)$ guarantees, that $u$ is a 
strictly convex function, see the definition of 
convexity for $M$ below. 
The strictly convex hypersurface $M$ evolves such that
its velocity in direction of the upwards pointing unit
normal vector $\nu$ is determined by
a given smooth positive function $f:\ol{\Omega}\to\R$
and a curvature 
function $F$ of the class $\Kstar$ defined below.
We remark that this class of curvature functions
contains especially the Gau{\ss} curvature.
The curvature function $F$ is evaluated at the vector
$(\kappa_i(X))$ the components of which are the principal
curvatures of $M$ at $X\in M$, $f$ is evaluated at $X$
where the $(n+1)$-th component of $X$ is ignored. The
image of the normal of $M$, $\nu(M)$, coincides with the
image of the normal of the smooth strictly convex 
hypersurface $M_0=\graph\left.(-u_0)\right|_\Omega$ we
start with. Here $u_0\in C^\infty\left(\ol\Omega\right)$
is assumed to be uniformly strictly convex up to the boundary.
We will assume that the closure of 
$\nu(M_0)$ is a geodesically strictly convex subset of
the unit sphere $S^n$ and contained in $S^n\cap\{x^{n+1}>0\}$.

From the definition of the unit normal $\nu$ of $M$ it 
follows that prescribing $\nu(M)=\nu(M_0)$ is equivalent
to prescribing $Du(\Omega)=Du_0(\Omega)=:\Omega^*$, where 
$\Omega^*$ is a strictly convex subset of $\R^n$. 
Thus we consider
a flow equation subject to a second boundary value 
condition. 

Under the assumptions stated above, we obtain the following
main theorem.
\begin{theorem}\label{main thm}
The initial value problem (\ref{flow eqn}) admits a
convex solution $M(t)=\graph\left.(-u(\cdot,t))\right|_{\Omega}$
that exists for all times $t\ge0$ and converges smoothly
to a translating solution $M^\infty(t)
=\graph\left.\left(-u^\infty(\cdot,t)\right)\right|_{\Omega}$
of the flow equation
\begin{equation}\label{flow}
\left\{\begin{array}{r@{~=~}l}
\dot X&-(\log F-\log f)\nu,\\[0.3ex]
\nu(M)&\nu(M_0),
\end{array}\right.
\end{equation}
i.\ e.\ there exists $v^\infty\in\R$ such that
$u^\infty(x,t)=u^\infty(x,0)+v^\infty\cdot t$.
Up to additive constants, the translating solution
is independent of the choice of $M_0$, but depends 
on $\nu(M_0)$, $F$, $f$ and $\Omega$.
 The function $u$ is smooth for $t>0$, $u\in C^\infty\left(
\ol\Omega\times(0,\infty)\right)$, $D^2u$ is positive 
definite up to the boundary, and $u$, $Du$,
$D^2u$, $\dot u$ are continuous up to $t=0$.
\end{theorem}

We mention some similar papers. In \cite{awcvp1994} the
authors study translating solutions for the mean curvature
flow whereas flow equations are considered in \cite{cwaih2000,
cgium2000, cgjdg1996} to prove existence for elliptic problems.
Flows with boundary conditions are studied in \cite{ghjde1989,
ilaia1997, osks2001}. Elliptic Hessian equations with
Neumann and oblique boundary conditions are solved
in \cite {ltu, jucvp1998}, the second boundary value
problem is considered in \cite{ju Hessian, jucj1997}
for Hessian and in \cite{ju Weingarten} for curvature
equations. 

Our paper is organized as follows. In Section~\ref{preliminaries}
we describe our differential-geometric notations, introduce
a class of curvature functions, rewrite our evolution equation
in non-parametric form, define the Legendre transformation,
state some properties of the curvature functions introduced,
and describe the effects of compatibility conditions to
solutions for short time intervals. We show that our boundary
condition is strictly oblique and prove lower order estimates
in Section~\ref{oblique est}. In Section~\ref{geom evol eqns}
we derive geometric evolution equations needed for the
$C^2$-a priori estimates proved in Section~\ref{C2 estimates}
and obtain an estimate for some terms in the evolution equations.
Then, we prove our main theorem in Section~\ref{proof main thm}
and conclude with some remarks on space-like hypersurfaces
in Minkowski space, convergence to
hypersurfaces of prescribed curvature 
and on translating solutions
for Hessian equations.

We wish to thank J\"urgen Jost and the Max Planck Institute
for Mathematics in the Sciences, Leipzig/Germany, for 
support. We acknowledge that part of this paper was written 
while we were visiting Shing-Tung Yau at Harvard University,
Cambridge/Massachusetts/USA, 
supported by the Alexander von Humboldt foundation. We also 
wish to thank the referee, especially for the suggestion of a
new barrier function.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}\label{preliminaries}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Geometric notations}

The notation used is very similar to \cite{cgjdg1996}.
Having fixed coordinate systems in $\R^{n+1}$ and $M$, where
we assume that the coordinates in $\R^{n+1}$ are Euclidean, we 
use Greek indices running from $0$ to $n$ to denote
components of geometric quantities defined in $\R^{n+1}$
and Latin indices starting at $1$ for quantities related
to the hypersurface $M$. Lower and upper indices refer to 
covariant and contravariant transformation properties,
respectively. We use the Einstein summation convention. 
Covariant derivatives are indicated by (additional) indices,
sometimes preceded by a semicolon for greater clarity, 
whereas a comma indicates partial derivatives, so we have
$$X^\alpha_{ij}=X^\alpha_{,ij}-\Gamma^k_{ij}X^\alpha_k,$$
where $\left(\Gamma^k_{ij}\right)$ denotes the Christoffel
symbols of $M$. These derivatives of the embedding vector
$X$ of $M$ are related to the second fundamental form $(h_{ij})$
and to the upwards pointing unit normal $(\nu^\alpha)$
of $M$ by the Gau{\ss} formula
$$X^\alpha_{ij}=-h_{ij}\nu^\alpha.$$
Using $M=\graph (-u)$, i.\ e.\ $X=(\cdot,-u)$, 
partial derivatives and lifting indices
with respect to the Kronecker-delta, we see that $\nu$ is given by
$$(\nu^\alpha)=\frac{1}{v}\left(1,u^i\right),\quad
v=\sqrt{1+u^iu_i}.$$
Covariant derivatives of $\nu$ can be expressed by the
Weingarten equation
$$\nu^\alpha_i=h^k_iX^\alpha_k,$$
where we lifted the index of the second fundamental form
with respect to $\left(g^{ij}\right)$, the inverse of the
induced metric $(g_{ij})$ of $M$,
$$g_{ij}=\delta_{ij}+u_iu_j,\quad g^{ij}=\delta^{ij}-
\frac{u^iu^j}{v^2}.$$
If not stated otherwise we will lift and lower indices
with respect to the induced metric when we use
covariant derivatives and with respect to the Kronecker-delta
if we use partial derivatives. The Codazzi equation 
-- together with the symmetry of the second fundamental
form -- states
that $h_{ij;\,k}$ is unchanged under permutations of the
indices. The Gau{\ss} equation gives the Riemannian
curvature tensor $(R_{ijkl})$ of $M$
$$R_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk},$$
used in the Ricci identity which we mention only for
the second fundamental form
$$h_{ik;lj}=h_{ik;jl}+h^a_kR_{ailj}+h^a_iR_{aklj}.$$
From the $0$-th component of the Gau{\ss} formula we obtain
\begin{equation}\label{h def}
\frac{1}{v}h_{ij}=u_{;ij},
\end{equation}
so $M$ is strictly convex if $u_{;ij}$ is positive definite.
Calculating $u_{;ij}=u_{,ij}-\Gamma^k_{ij}u_k=\frac{1}{v^2}u_{,ij}$,
we see that the convexity of $M$ is equivalent to the convexity
of $u$. Of course, the function $u$ is called convex if
$u(\cdot,t)$ is convex for all $t$. 

In what follows we rewrite our evolution equation as follows
$$\dot X=-(\log F-\log f)\nu\equiv-\left(\hat F-\hat f\right)
\nu.$$

Sometimes it will be convenient to work with indices that
indicate partial derivatives. We will point out this
in the respective sections. In contrast to the lifting of
indices as mentioned above, $\left(u^{ij}\right)$ denotes the 
inverse of $(u_{,ij})$. We also wish to introduce the abbreviation
$u_\nu=u_i\nu^i$ for a vector $\nu$. The letter $c$ is 
used to denote constants. These constants are positive
estimated quantities that may change its value from line
to line. Inequalities remain valid if a constant $c$ on
the ``right-hand side'' is enlarged.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Curvature functions}\label{curv fct}
We introduce some classes of curvature functions similar
to \cite{cgium2000, osmz}. A slightly different
class of curvature functions is considered in \cite{ju
Weingarten}. Our choice of the class of curvature functions
used in our main theorem is not the most general choice
possible. Instead we preferred a choice that corresponds 
to the examples of curvature functions we know for which
such a theorem holds. 

Let $\Gamma_+\subset\R{}^n$ be the open positive cone and
$F\in C^{\infty}(\Gamma_+)\cap C^0\left(\ol{\Gamma}_+\right)$
a symmetric function satisfying the condition
$$F_i=\fracp{F}{\kappa^i}>0;$$
then, $F$ can also be viewed as a function defined on the space
of symmetric, positive definite matrices $Sym^+(n)$, for, let
$(h_{ij})\in Sym^+(n)$ with eigenvalues $\kappa_i$, $1\le i\le n$,
then define $F$ on $Sym^+(n)$ by
$$F(h_{ij})=F(\kappa_i).$$
We have
$F\in C^{\infty}\left(Sym^+\right)\cap C^0\left(\ol{Sym^+}\right)$.
If we define
$$F^{ij}=\fracp{F}{h_{ij}},$$
then we get in an appropriate coordinate system
$$F^{ij}\xi_i\xi_j=\fracp{F}{\kappa_i}\left|\xi^i\right|^2
  \quad\forall\,\xi\in\R{}^n,$$
and $F^{ij}$ is diagonal, if $h_{ij}$ is diagonal.
We define furthermore
$$F^{ij,kl}=\frac{\partial^2 F}{\partial h_{ij}\partial h_{kl}}.$$

\begin{definition}\label{DefinitionClassK}
A curvature function $F$ is said to be of the class $(K)$, if
\begin{equation}\label{K differentiability}
F\in C^{\infty}(\Gamma_+)\cap C^0\left(\ol{\Gamma}_+\right),
\end{equation}
\begin{equation}\label{K symmetry}
F\mbox{ is symmetric,}
\end{equation}
\begin{equation*}%\label{K homogeneity}
F\mbox{ is positive homogeneous of degree }d_0>0,
\end{equation*}
\begin{equation}\label{K elliptic}
F_i=\fracp{F}{\kappa_i}>0\quad\mbox{in }\Gamma_+,
\end{equation}
\begin{equation}\label{K vanishes}
\left.F\right|_{\partial\Gamma_+}=0,
\end{equation}
and
\begin{equation*}%\label{DefinitionClassKPropertyK}
  F^{ij,kl}\eta_{ij}\eta_{kl}\le F^{-1}
  \left(F^{ij}\eta_{ij}\right)^2-F^{ik}\tilde h^{jl}\eta_{ij}
  \eta_{kl}\quad\forall\,\eta\in Sym,
\end{equation*}
where
$\left(\tilde h^{ij}\right)$ denotes the
inverse of $(h_{ij})$, or, equivalently,
if we set $\hat F=\log F$,
\begin{equation}\label{concave inequ}
\hat F^{ij,kl}\eta_{ij}\eta_{kl}\le
  -\hat F^{ik}\tilde h^{jl}\eta_{ij}
  \eta_{kl}\quad\forall\,\eta\in Sym,
\end{equation}
where $F$ is evaluated at $(h_{ij})$.

If $F$ satisfies
\begin{equation}\label{* inequ}
\exists\,\varepsilon_0>0:\quad
  \varepsilon_0 FH\equiv\varepsilon_0 F\,\tr h_i^j\le
  F^{ij}h_{ik}h^k_j
\end{equation}
for any $(h_{ij})\in Sym^+$, where the index is lifted by means of
the Kronecker-Delta, then we indicate this by using an
additional star, $F\in(K^\star)$.

The class of curvature functions $F$ which fulfill, instead of the
homogeneity condition, the following weaker assumption
\begin{equation}\label{rel hom}
\exists\,\delta_0>0:\quad 0<\frac{1}{\delta_0}F\le
  \sum\limits_iF_i\kappa_i\le\delta_0F
\end{equation}
is denoted by an additional tilde, $F\in\left(\tilde K\right)$
or $F\in\left(\tilde K^\star\right)$.

A curvature function $F$ which satisfies for any
$\varepsilon>0$
$${F(\varepsilon,\ldots,\varepsilon,R)\to+\infty,\quad
  \mbox{as }R\to+\infty,}$$
or equivalently
$${F(1,\ldots,1,R)\to+\infty,\quad\mbox{as }R\to+\infty,}$$
in the homogeneous case, a condition similar to an
assumption in \cite{cnsv}, is said to be of the class
$(CNS)$.
\end{definition}

\begin{example}
We mention examples of curvature functions of the class
$\left(\tilde K^\star\right)$ as given in \cite{cgium2000,
osmz}.

Let $H_k$ be the $k$-th elementary symmetric polynomial,
\begin{eqnarray}
H_k(\kappa_i)&:=&\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,\\
  \sigma_k&:=&\left(H_k\right)^\frac{1}{k}\nonumber
\end{eqnarray}
the respective curvature function homogeneous of degree $1$
and define furthermore
\begin{equation}\nonumber
\tilde\sigma_k(\kappa_i):=\frac{1}{\sigma_k
  \left(\kappa_i^{-1}\right)}\equiv \left(S_{n,n-k}
  \right)^\frac{1}{k}.
\end{equation}

The functions $S_{n,k}$ belong to the class $(K)$ for
$1\le k\le n-1$ and $H_n$ belongs to the class
$\left(K^\star\right)$.

Furthermore, see \cite{cgium2000},
\begin{equation}\label{K times example}
F:=H_n^{a_0}\cdot\prod\limits_{i=1}^{N}F_{(i)}^{a_i},
\quad a_i>0,
\end{equation}
belongs to the class $\left(\tilde K^\star\right)$
provided $F_{(i)}\in\left(\tilde K\right)$, and we may even allow
$F_{(i)}\not=0$ on $\partial\Gamma_+$.

An additional construction gives inhomogeneous
examples \cite{osmz}.
Let $F$ be as in (\ref{K times example}),
$\eta\in C^{\infty}\left(\R_{\ge0}\right)$
and $c_\eta>0$ such that
\begin{equation}\nonumber
0<\frac{1}{c_\eta}\le\eta\le c_\eta,\quad\eta'\le0,
\end{equation}
then
\begin{equation}\nonumber
\tilde F(\kappa_i):=F\left(\exp\left(\int\limits_1^{\kappa_i}
\frac{\eta(\tau)}{\tau}\,d\tau\right)\right)
\end{equation}
belongs to the class $\left(\tilde K^\star\right)$.
\end{example}

The considerations above remain applicable if we 
evaluate $F$ in what follows at the eigenvalues
$(\kappa_i)$ of the second fundamental form $(h_{ij})$
with respect to the metric $(g_{ij})$, i.\ e.\ $\kappa$ 
is an eigenvalue if there exists $\xi\in\R^n\setminus\{0\}$
such that
$$\kappa\cdot g_{ij}\xi^j=h_{ij}\xi^j.$$
Then
$$F^{ij}=\fracp{F}{h_{ij}}$$
is a symmetric contravariant tensor of second order.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Non-parametric flow equation}
Our boundary condition guarantees that we can represent our
solution as $\graph u$. Now we will derive a parabolic 
evolution equation for $u$ equivalent to 
$$\dot X=-\left(\hat F-\hat f\right)\nu.$$
Therefore we choose local coordinates $\left(x^i\right)$ of $\R^n$
and obtain
\begin{eqnarray*}
\frac{d}{dt}X^0&=&-\frac{d}{dt}u\left(X^i(x,t),t\right)
=-\fracp{u}{t}-u_i\dot X^i=-\left(\hat F-\hat f\right)
\frac{1}{v},\\
\frac{d}{dt}X^i&=&-\left(\hat F-\hat f\right)\frac{u^i}{v},
\quad 1\le i\le n,
\end{eqnarray*}
where we used the definition of $\nu$ and
(\ref{flow eqn}). Combining these
equations yields
$$\frac{d}{dt}u(x,t)=\fracp{u}{t}=v\left(\hat F-\hat f
\right).$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Legendre transformation}\label{leg trans}
In this section we use indices to denote partial derivatives
and ignore our convention that upper and lower indices
correspond to contravariant and covariant quantities,
respectively.

The Legendre-transformation of $u:\Omega\times[0,T)\to\R$,
$u^*:\Omega^*\times[0,T)\to\R$, is defined by
$$u^*(y,t):=x^iu_i(x,t)-u(x,t)\equiv x^iy_i-u,
\quad y^i=u^i(x,t).$$
We look for an evolution equation for $u^*$. From the
definition of $u^*$ we get
$$\dot u^*=-\dot u,\quad\fracp{u^*}{y^k}=x_k,\quad
\frac{\partial^2u^*}{\partial y^k\partial y^l}=
\left((D^2u)^{-1}\right)_{kl}\equiv u^{kl},$$
where $y$ is considered as a time independent variable.
We use $\left(\sqrt{g}^j_i\right)$ and $\left(\sqrt{g^{-1}}^j_i
\right)$ to denote the square roots of $(g_{ij})$ and $\left(g^{ij}
\right)$, respectively,
 which are positive definite symmetric matrices such that 
$\sqrt{g}^j_i\sqrt{g}^k_j\delta_{kl}=g_{il}$ and
$\sqrt{g^{-1}}^i_j\sqrt{g^{-1}}^j_k\delta^{kl}=g^{il}$,
explicitly
$$\sqrt{g}^j_i=\delta^j_i+\frac{u_iu^j}{1+v},\quad
\sqrt{g^{-1}}^j_i=\delta^j_i-\frac{u_iu^j}{v(1+v)}.$$
Then, following \cite{nibcp1996}, the principal curvatures
$\kappa_i$, $1\le i\le k$, are the eigenvalues of the 
matrix $(a_{ij})$, where
$$a_{ij}=\sqrt{g^{-1}}^k_i\frac{u_{kl}}{v}\sqrt{g^{-1}}^l_j.$$
We may consider $\sqrt{g}$ as a function of $y$ and set
$$a^*_{ij}=v\sqrt{g}^i_ku^{kl}\sqrt{g}^j_l
=\sqrt{1+|y|^2}\sqrt{g}^k_i(y)u^*_{kl}\sqrt{g}^l_j(y).$$
Then the eigenvalues of $a^*_{ij}$ are given by 
$\frac{1}{\kappa_i}$, $1\le i\le n$.
We set for $\kappa\in\Gamma_+$
$$F^*(\kappa_i):=\frac{1}{F\left(\frac{1}{\kappa_i}\right)}$$
and 
$$f^*=\frac{1}{f}.$$
Thus we obtain the following evolution equation for $u^*$
$$\left\{\begin{array}{r@{~=~}l}
\dot u^*&\sqrt{1+|y|^2}\left(\log F^*(a^*_{ij})-\log f^*
(Du^*)\right)\quad\mbox{in~}\Omega^*,\\
Du^*(\Omega^*)&\Omega,
\end{array}\right.$$
where $F^*$ is evaluated at the eigenvalues of $a^*_{ij}$.
For later use we differentiate this flow equation using
the index $k$ for derivatives with respect to $y^k$
\begin{eqnarray}\label{dot uk*}
\dot u^*_k&=&\frac{y_k}{\sqrt{1+|y|^2}}\left(\hat F^*
(a^*_{ij})-\hat f^*(Du^*)\right)\\
&&{}+\sqrt{1+|y|^2}
\left(\hat F^*_{u^*_{ij}}u^*_{ijk}+\hat F^*_{y^k}
-\hat f^*_{q_i}u^*_{ik}\right).\nonumber
\end{eqnarray}
We compute $\hat F^*_{u^*_{ij}}$ and $\hat F^*_{y^k}$
explicitly, 
$$\hat F^*_{u^*_{ij}}=\hat F^*_{a^*_{kl}}\sqrt{1+|y|^2}
\sqrt{g}^i_k\sqrt{g}^j_l$$
and
$$\hat F^*_{y^k}=\hat F^*_{a^*_{ij}}\left(\left(\sqrt{1+
|y|^2}\sqrt{g}^a_i\right)_ku^*_{ab}\sqrt{g}^b_j
+\sqrt{1+|y|^2}\sqrt{g}^a_iu^*_{ab}\sqrt{g}^b_{j,k}\right).$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Properties of curvature functions}
Important properties of the class $\left(\tilde K^\star\right)$
for the a priori estimates of
the second derivatives of $u$ at the boundary are
stated in the following lemmata.

\begin{lemma}\label{three equivalences}
Let $F\in\left(\tilde K^\star\right)$, then for
fixed $\varepsilon>0$
$$F(\varepsilon,\,\ldots\,,\varepsilon,R)\to\infty
\quad\mbox{as~}R\to\infty,$$
i.\ e. $\left(\tilde K^\star\right)\subset\left(\tilde K\right)\cap
(CNS)$, moreover, when $F\in\left(\tilde K\right)\cap(CNS)$,
$0<\frac{1}{c}\le F\le c$, and
\begin{equation}\nonumber
0<\kappa_1\le\ldots\le\kappa_n,
\end{equation}
then the following three conditions are equivalent
\begin{equation*}
\kappa_1\to0,\quad
\kappa_n\to\infty,\quad
\mbox{and}\quad
{\rm tr\,}F^{ij}\to\infty.
\end{equation*}
\end{lemma}
\begin{proof}
We refer to \cite{osmz}.
\end{proof}

For the dual functions we have a similar lemma.
\begin{lemma}\label{three equivalences for F*}
Let $F\in\left(\tilde K^\star\right)$,
\begin{equation}\nonumber
0<\kappa_1\le\ldots\le\kappa_n,
\end{equation}
and $0<\frac{1}{c}\le F\le c$. Then the following
three conditions are equivalent
$$\kappa_1\to0,\quad
\kappa_n\to\infty,\quad
\mbox{and}\quad
{\rm tr\,}F^{*ij}\to\infty.$$
\end{lemma}
\begin{proof}
We have $F_1\ge\ldots\ge F_n>0$, see \cite{cgmz1997,
ju Hessian}, so we get in view of the definition
of $F^*$
$$F^*_i(\kappa_1,\ldots,\kappa_n)=
\frac{F_i\left(\frac{1}{\kappa_i}\right)}{F^2}
\cdot\frac{1}{\kappa_i^2}.$$
Thus $F^*_1\to\infty$ as $\kappa_1\to0$ gives one implication, 
and $\kappa_n\to\infty$ is equivalent to $\kappa_1\to0$
in view of Lemma \ref{three equivalences}.
To get $\tr F^{*ij}\to\infty$, $\kappa$ has to leave any
compact subset of $\Gamma_+$.
\end{proof}

\begin{lemma}\label{properties of F*...}
Let $F\in\left(\tilde K\right)\cap(CNS)$.
Then $F^*$ as defined above satisfies
(\ref{K differentiability}), (\ref{K symmetry}),
(\ref{K elliptic}), (\ref{K vanishes}) and
$F^*\in(CNS)$. For $F=(S_{n,k})^{\frac{1}{n-k}}$, $1\le
k\le n-1$, and obviously, see Lemma \ref{three equivalences},
also for $F\in\left(\tilde K\right)
\cap(CNS)$, we have for any $\varepsilon>0$
\begin{equation}
\sum\limits_iF_i\kappa_i^2\le(c(\varepsilon)+\varepsilon\cdot
|\kappa|)\cdot\sum\limits_iF_i,
\end{equation}
provided that $0<\frac 1c\le F\le c$.
\end{lemma}
\begin{proof}
See \cite{ju Hessian}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Shorttime existence and compatibility conditions}

In this section we use partial derivatives.
In the introduction,
we have rewritten our boundary condition $\nu(M)=\nu(M_0)$
equivalently as $Du(\Omega)=\Omega^*$. Now, we take a smooth
strictly concave function $h:\R^n\to\R$ such that
$h=0$ and $|\nabla h|=1$ on $\partial\Omega^*$.
In what follows we use $h_{p_k}$ instead of $h_k$ as $h$
will be evaluated by using the gradient of a function.
For smooth strictly convex functions $u$, our boundary 
condition is equivalent to $h(Du)=0$ on $\partial\Omega$.

We will derive compatibility conditions fulfilled by a smooth
solution $u:\ol{\Omega}\times[0,T)\to\R$ and show then
how compatibility conditions affect the regularity of $u$
at $t=0$. We take a solution $u$, smooth up to $t=0$, and
compute time derivatives of our boundary condition,
$$\left.\left(\frac{d}{dt}\right)^mh(Du)\right|_{t=0}=0
\quad\mbox{on~}\partial\Omega,\quad m\in\N.$$
For fixed $m$, we call this equation 
the compatibility condition of order $m$.
For $m=0$ we get back our boundary condition. 
If $m\ge1$, we can substitute time derivatives
of $u$, $Du$, \ldots, inductively by using
$\dot u=v\left(\hat F-\hat f\right)$ and derivatives
of this equation. Thus we can express compatibility
conditions of any order so that they contain only 
spatial derivatives of $u$ at $t=0$. These necessary
conditions for smoothness of a solution of (\ref{flow eqn})
at $t=0$ are also sufficient for smoothness, more precisely, let 
$M_0=\graph u_0$ satisfy the compatibility conditions of
$m$-th order for $0\le m\le m_0$. Later-on we will prove
that our boundary condition is strictly oblique, so
we deduce from Theorem~5.3, p.~320 \cite{lady}, and
the implicit function theorem, see also \cite{cgln2001},
that there exists a solution of our initial value
problem (\ref{flow eqn}) on a maximal time interval
$[0,T)$, $T>0$. This solution is smooth for $t\in(0,T)$ and
has continuous derivatives up to $2(m_0+1)$-th order at
$t=0$, where time derivatives have to be counted twice.

As usual, longtime existence follows from shorttime 
existence and uniform a priori estimates as follows.
Assume that we have for $T>t\ge\epsilon>0$
estimates for the function $u$ of the form
$$\Vert u\Vert_{C^k\left(\ol{\Omega}\times\left(\frac{t}{2},t
\right)\right)}
\le c_k\cdot (1+t),\quad k\in\N,$$
where it would be sufficient to have a locally bounded 
function -- defined for $t>0$ -- of $t$ on the 
right-hand side. We assume that $[0,T)$,
$T<\infty$, is the maximal
time interval, where our solution exists. Then the a priori
estimates guarantee, that we can extend our solution to
$[0,T]$. As we have a smooth solution, it satisfies the 
compatibility conditions of any order at $t=T$, so 
applying our considerations above, we get a solution on a
time interval $[0,T+\varepsilon)$ for some $\varepsilon>0$,
which is smooth for $t\in(0,T+\varepsilon)$ contradicting
the maximality of $T$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Obliqueness and lower order estimates}\label{oblique est}

In this section we use indices to denote partial 
derivatives and $\nu$ is the inner unit normal vector to
$\partial\Omega$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Strict obliqueness}

\begin{lemma}\label{obl lem}
As long as a solution of (\ref{flow eqn}) exists,
our boundary condition is strictly oblique, i.\ e.\
\begin{equation}\label{strict obliqueness}
\langle\nu(x),\nu^*(Du(x,t))\rangle>0,
\quad x\in\partial\Omega,
\end{equation}
where $\nu$ and $\nu^*$ denote the inner unit normals of
$\Omega$ and $\Omega^*$, respectively.
\end{lemma}
For a similar result we refer to \cite{ju Weingarten}.
\begin{proof}
To prove (\ref{strict obliqueness}) we use
$$\nu^i(x)\cdot\nu^*_i(Du(x,t))=\nu^i\cdot h_{p_i}(Du(x,t)).$$
As $h(Du)$ is positive in $\Omega$ and vanishes on $\partial
\Omega$, we get on $\partial\Omega$ for $\tau$ orthogonal
to $\nu$
\begin{equation}\label{h(Du)=0 differentiated}
h_{p_k}u_{k\tau}=0,\quad h_{p_k}u_{k\nu}\ge0.
\end{equation}
Thus we see from
\begin{equation}\label{degenerate obliqueness}
h_{p_k}\nu^k=h_{p_k}u_{ki}u^{ij}\nu_j=h_{p_k}u_{k\nu}
\cdot u^{\nu\nu}\ge 0
\end{equation}
that the quantity whose positivity we wish to show is at
least nonnegative. 
\par
We compute in view of (\ref{h(Du)=0 differentiated})
and (\ref{degenerate obliqueness}) on $\partial\Omega$
\begin{eqnarray*}
\left(h_{p_k}\nu^k\right)^2
&=&u^{\nu\nu}h_{p_k}u_{k\nu}u^{\nu\nu}
u_{\nu l}h_{p_l}\\
&=&u^{\nu\nu}h_{p_k}u_{ki}u^{ij}u_{jl}h_{p_l}\\
&=&u^{\nu\nu}u_{kl}h_{p_k}h_{p_l},
\end{eqnarray*}
so we deduce the positivity of the quantity
considered.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\dot u$- and $C^0$-estimates}

We define the function
$$r:=(\dot u)^2$$
and consider $F(\kappa_i)$ as a function of $(Du, D^2u)$,
$F=F(Du,D^2u)$. Calculations similar to those
in Section \ref{leg trans}
show that $\left(F_{u_{ij}}\right)$ is positive definite.
We get
$$\ddot u=v\left(\hat F_{u_{ij}}\dot u_{ij}+
\hat F_{p_i}\dot u_i\right)+\frac{u^i\dot u_i}{\sqrt{1+|Du|^2}}
\left(\hat F-\hat f\right).$$
These preparations allow to deduce the following
parabolic evolution equation for $r$
$$\dot r-v\hat F_{u_{ij}}r_{ij}=-2v\hat F_{u_{ij}}\dot u_i\dot u_j
+v\hat F_{p_i}r_i+\frac{u^ir_i}{\sqrt{1+|Du|^2}}
\left(\hat F-\hat f\right).$$

\begin{lemma}
As long as a smooth solution of (\ref{flow eqn}) exists, we
obtain the estimate
$$\min\left\{\min\limits_{t=0}\dot u,0\right\}
\le\dot u\le\max\left\{\max\limits_{t=0}\dot u,0\right\}.$$
\end{lemma}
\begin{proof}
We repeat the proof given in \cite{osks2001}.
If $(\dot u)^2$ admits a local maximum at $x\in\partial
\Omega$ for some positive time, we differentiate our boundary
condition and get there
$$h_{p_k}\dot u_k=0.$$
As $h_{p_k}$ is strictly oblique, this contradicts the Hopf
maximum principle unless $\dot u$ is constant.
On the other hand, the evolution equation for $r$
implies that
$$\dot r-v\hat F_{u_{ij}}r_{ij}\le
v\hat F_{p_i}r_i+\frac{u^ir_i}{\sqrt{1+|Du|^2}}
\left(\hat F-\hat f\right).$$
This excludes a strictly increasing local maximum of $(\dot u)^2$
in $\Omega\times(0,T)$, where we assume that a solution 
exists, so we get the claimed inequality.
\end{proof}

\begin{corollary}\label{C0 est}
As long as a smooth solution of (\ref{flow eqn}) exists, we 
get
$$\Vert u(\cdot,t)\Vert_{C^0\left(\ol{\Omega}\right)}
\le\Vert u(\cdot,0)\Vert_{C^0\left(\ol{\Omega}\right)}
+t\cdot\Vert\dot u(\cdot,0)\Vert_{C^0\left(\ol{\Omega}\right)}$$
and $\hat F$, $\hat F-\hat f$ are uniformly a priori bounded.
\end{corollary}

\begin{remark}
The boundary condition $Du(\Omega)=\Omega^*$ ensures that
$|Du|$ remains uniformly bounded during the flow. 
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Strict obliqueness estimates}

The purpose of this section is to quantify the strict
obliqueness of our boundary condition. 
We adapt the argument of \cite{ju Weingarten}.

\begin{lemma}
For a smooth solution of (\ref{flow eqn}) we have the
strict obliqueness estimate
$$\langle\nu(x),\nu^*(Du(x,t))\rangle\ge\frac{1}{c}>0,
\quad x\in\partial\Omega,$$
where $\nu$ and $\nu^*$ denote the inner unit normals
of $\Omega$ and $\Omega^*$, respectively. The positive
lower bound is independent of time.
\end{lemma}
\begin{proof}
We fix a time interval $(0,T]$, where a smooth solution
of our flow (\ref{flow eqn}) exists and prove that there
exists a positive lower bound for $h_{p_k}\nu^k$ for
$(x,t)\in\partial\Omega\times[0,T]$ which is independent
of $T$. To establish this positive lower bound,
we choose $(x_0,t_0)\in\partial\Omega\times[0,T]$ such
that $h_{p_k}\nu^k$ is minimal there. As we have a positive
lower bound for $h_{p_k}\nu^k$ on $\partial\Omega\times
\{0\}$, we may assume that $t_0>0$. Further on, we may assume
that $\nu(x_0)=e_n$ and extend $\nu$ smoothly to a tubular
neighborhood of $\partial\Omega$ such that in the matrix
sense
\begin{equation}\label{Omega strictly convex}
D_k\nu^l\equiv\nu^l_k\le-\frac{1}{c_1}\delta^l_k
\end{equation}
there for a positive constant $c_1$. For a positive constant
$A$ to be chosen below we define
$$w=h_{p_k}\nu^k+Ah(Du).$$
The function $\left.w\right|_{\partial\Omega\times(0,T]}$
attains its minimum over $\partial\Omega\times(0,T]$ in
$(x_0,t_0)$, so we deduce there
\begin{eqnarray}\label{tangential derivatives vanish}
0&=&w_r=h_{p_np_k}u_{kr}+h_{p_k}\nu^k_r+Ah_{p_k}u_{kr},
\quad 1\le r\le n-1,\\
\label{time derivative negative}
0&\ge&\dot w.
\end{eqnarray}
We assume for a moment that there holds
\begin{equation}\label{bound for v_n}
w_n(x_0,t_0)\ge-c(A),
\end{equation}
show that this estimate yields a positive lower bound for
$u_{kl}h_{p_k}h_{p_l}$ and prove (\ref{bound for v_n})
afterwards. Then the lemma follows from the calculations
in the proof of Lemma \ref{obl lem} and
from a positive lower bound for $u^{\nu\nu}$.
\par
We rewrite (\ref{bound for v_n}) as
$$h_{p_np_l}u_{ln}+h_{p_k}\nu^k_n+Ah_{p_k}u_{kn}\ge-c(A).$$
Multiplying this with $h_{p_n}$ and adding (\ref{tangential
derivatives vanish}) multiplied with $h_{p_r}$ we obtain
at $(x_0,t_0)$
$$Au_{kl}h_{p_k}h_{p_l}\ge-c(A)h_{p_n}-h_{p_k}\nu^k_lh_{p_l}
-h_{p_k}h_{p_np_l}u_{lk}.$$
Using (\ref{h(Du)=0 differentiated}), the concavity of $h$
and (\ref{Omega strictly convex}), we get there
$$Au_{kl}h_{p_k}h_{p_l}\ge-c(A)h_{p_n}+\frac{1}{c_1}$$
as $|\nabla h|=1$ on $\partial\Omega^*$. We may assume that
the right-hand side of the inequality above is positive as
otherwise $h_{p_n}=h_{p_k}\nu^k$ is bounded from below.
Thus we deduce a positive lower bound for $u_{kl}h_{p_k}
h_{p_l}$.
\par
We now sketch the proof of (\ref{bound for v_n}). As
for a similar proof with more details we refer to
\cite{osks2001}. We differentiate our flow equation
$\dot u=v\left(\hat F-\hat f\right)$ and obtain
\begin{equation}\label{dot uk}
\dot u_k=v\left(\hat F_{u_{ij}}u_{ijk}+\hat F_{p_i}
u_{ik}-\hat f_k\right)+\left(\hat F-\hat f\right)
v_{p_i}u_{ik}.
\end{equation}
This is a motivation to introduce the following linear
parabolic differential operator $L$ by
$$L\tilde w:=-\dot{\tilde w}+v\hat F_{u_{ij}}\tilde w_{ij}
+v\hat F_{p_i}\tilde w_i+\left(\hat F-\hat f\right)
v_{p_i}\tilde w_i.$$
We remark that the chain rule and (\ref{rel hom})
show that $\hat F_{p_i}$ is bounded, the chain rule
and Lemma \ref{three equivalences} give a positive lower
bound for $\tr\hat F_{u_{ij}}\equiv\hat F_{u_{ij}}
\delta_{ij}$.
Direct calculations give for $A$ sufficiently large
\begin{eqnarray*}
Lw&\le&v\hat F_{u_{ij}}u_{li}u_{mj}\nu^kh_{p_kp_lp_m}
+Av\hat F_{u_{ij}}h_{p_kp_l}u_{ki}u_{lj}
+c(A)\cdot\tr\hat F_{u_{ij}}\\
&\le&c(A)\cdot\tr\hat F_{u_{ij}}.
\end{eqnarray*}
As $\Omega$ is strictly convex, there exist $\mu\gg1$ and
$\varepsilon>0$ such that for $\vartheta:=d-\mu d^2$,
where $d=\dist(\cdot,\partial\Omega)$, we have near
$\partial\Omega$ in view of Lemma \ref{three equivalences}
\begin{equation}\label{estimate for theta}
L\vartheta\le-\varepsilon\cdot\tr\hat F_{u_{ij}}.
\end{equation}
The proof of this inequality is omitted here as it is
carried out in \cite{osks2001} and in Lemma \ref{cov est}
in similar situations.
We consider $\vartheta$ only in $\Omega_\delta:=
\Omega\cap B_\delta(x_0)$, where $\delta>0$ is chosen
so small that $\vartheta$ is smooth and nonnegative
there and the above inequality holds.
We fix an affine linear function $l$ such that 
$l(x_0)=0$ and $w-w(x_0,t_0)+l\ge0$ for $t=0$.
As $w$ is bounded and attains its minimum over
$\partial\Omega\times[0,T]$ in $(x_0,t_0)$ we find
$C\gg B\gg1$ such that the function
$$\Theta:=C\cdot\vartheta+B\cdot|x-x_0|^2+w-w(x_0,t_0)+l$$
satisfies
$$\left\{\begin{array}{rcll}
\Theta&\ge&0&
\mbox{on~}(\partial\Omega_\delta\times[0,T])\cup
(\Omega_{\delta}\times\{0\}),\\[0.3ex]
L\Theta&\le&0&
\mbox{in~}\Omega_\delta\times[0,T].
\end{array}\right.$$
Thus the maximum principle gives
$$(C\cdot\vartheta+w+l)_n(x_0,t_0)\ge0$$
as the function $C\cdot\vartheta+B\cdot|x-x_0|^2
+w-w(x_0,t_0)+l$
vanishes in $(x_0,t_0)$. This shows Inequality
(\ref{bound for v_n}).
\par
Similar to the argument above we extend $\nu^*$ smoothly
to a tubular neighborhood of $\partial\Omega^*$ such that
$\nu^{*k}_i\le-\frac{1}{c}\delta^k_i$ in the matrix sense
and take $h^*$ as a smooth strictly concave function such
that $\{h^*=0\}=\partial\Omega$ and $|\nabla h^*|=1$ on
$\partial\Omega$. We define
$$w^*=h^*_{q_k}(Du^*)\nu^{*k}+Ah^*(Du^*)$$
and in view of (\ref{dot uk*}) we define furthermore
$$L^*\tilde w:=-\dot{\tilde w}
+v\hat F_{u^*_{ij}}\tilde w_{ij}-v\hat f^*_{q_i}\tilde w_i.$$
As before we obtain that $\left.w^*\right|_{\partial\Omega
\times[0,T]}$ is
positive. We fix $T>0$ and assume that $\left.w^*\right|_%
{\partial\Omega\times[0,T]}$ attains its minimum in $(x_0,
t_0)$. As we wish to establish a positive lower bound for
$w^*$ we may assume that $t_0>0$. Direct calculations and
Lemma \ref{three equivalences for F*}, which implies a 
positive lower bound for $\tr\hat F^*_{u^*_{ij}}$, give
for $A$ sufficiently large
\begin{equation}\label{*w est}
L^*w^*\le c_k(1+A)\left|\hat F^*_{y^k}\right|+\frac{1}{2}Av
\hat F^*_{u^*_{ij}}h^*_{q_kq_l}u^*_{ki}u^*_{lj}
+c(A)\cdot\tr\hat F^*_{u^*_{ij}}.
\end{equation}
Then
$$\hat F^*_{a^*_{ij}}=\frac{1}{v}\hat F^*_{u^*_{rs}}
\sqrt{g^{-1}}^r_i\sqrt{g^{-1}}^s_j$$
and Young's inequality imply for any $\varepsilon>0$
$$\left|\hat F^*_{y^k}\right|\le\varepsilon\hat F^*_{u^*_{rs}}
u^*_{ri}u^*_{sj}\delta^{ij}+\frac{c}{\varepsilon}
\tr\hat F^*_{u^*_{rs}}.$$
Combining this with (\ref{*w est}) gives
$$L^*w^*\le c(A)\cdot\tr\hat F^*_{u^*_{ij}}.$$
Now we can proceed as above, use Lemma \ref{three equivalences 
for F*} and get at $(x_0,t_0)$ an inequality of the form
\begin{equation}\label{*inequality}
Au^*_{kl}h^*_{q_k}h^*_{q_l}\ge-c(A)h^*_{q_k}\nu^{*k}
-\nu^{*l}_kh^*_{q_k}h^*_{q_l}.
\end{equation}
Since $h^*_{q_k}\nu^{*k}=\langle\nu^*,\nu\rangle$, we may
assume again that this quantity is small. The second term
on the right-hand side is bounded below by a positive
constant in view of the convexity of $\Omega^*$ and
$|\nabla h^*|=1$ on $\partial\Omega^*$, so we deduce that
$u^*_{kl}h^*_{q_k}h^*_{q_l}\ge\frac{1}{c}>0$. Using
$u^*_{kl}=u^{kl}$ and $h^*_{q_k}=\nu^k$ we obtain a positive
lower bound for $u^{\nu\nu}$ completing the strict obliqueness
estimate.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometric evolution equations}\label{geom evol eqns}
In this section we describe how geometric quantities
evolve during the flow. Proofs for these results in a
similar notation can be found in \cite{cgjdg1996}. We start
with the evolution equations for the metric
$$\dot g_{ij}=-2\left(\hat F-\hat f\right)h_{ij}$$
and for the unit normal of the hypersurface $M$
$$\dot\nu^\alpha=g^{ij}\left(\hat F-\hat f\right)_iX^\alpha_j.$$
For the second fundamental form of $M$ we state the evolution
equation both for the mixed and for the covariant form
\begin{eqnarray*}
\dot h^j_i&=&\left(\hat F-\hat f\right)_i{}^j+
\left(\hat F-\hat f\right)h^k_ih^j_k,\\
\dot h_{ij}&=&\left(\hat F-\hat f\right)_{ij}
-\left(\hat F-\hat f\right)h^k_ih_{kj}.
\end{eqnarray*}
Applying the chain rule to the term $\hat F-\hat f$ there
and interchanging covariant derivatives by means of the 
Codazzi equations and Ricci identities gives
\begin{eqnarray}\label{2ff evol}
\dot h^j_i-\hat F^{kl}h^j_{i;kl}
&=&\hat F^{kl}h_{kr}h^r_lh^j_i
-\hat F^{kl}h_{kl}h^r_ih^j_r
+\left(\hat F-\hat f\right)h^k_ih^j_k\\
&&{}+\hat F^{kl,rs}h_{kl;i}h_{rs;}{}^j
-\hat f_{\alpha\beta}X^\alpha_iX^\beta_kg^{kj}
+\hat f_\alpha\nu^\alpha h^j_i.\nonumber
\end{eqnarray}
For the scalar product $\tilde v$ of $\nu$ and $e_{n+1}=e_0$,
i.\ e.\ for $\tilde v\equiv\langle\nu,\eta\rangle=\nu^\alpha
\eta_\alpha$, where $(\eta_\alpha)=(1,0,\ldots,0)$, we
get the evolution equation
$$\dot{\tilde v}-\hat F^{ij}\tilde v_{ij}=
\hat F^{ij}h^k_ih_{kj}\tilde v
-\hat f_\beta X^\beta_iX^\alpha_j\eta_\alpha g^{ij}$$
and for $v=\tilde v^{-1}$ we get thus
$$\dot v-\hat F^{ij}v_{ij}=-v\hat F^{ij}h^k_ih_{kj}
+v^2\hat f_\beta X^\beta_iX^\alpha_j\eta_\alpha g^{ij}
-2\frac{1}{v}\hat F^{ij}v_iv_j.$$
For these two evolution equations we assumed Euclidean
coordinates in $\R^{n+1}$, so derivatives of $\eta_\alpha$
vanish. In the following we will always assume that
we have chosen Euclidean coordinates in $\R^{n+1}$.

The right-hand side of (\ref{2ff evol}) is a tensor 
with covariant index $i$ and contravariant index $j$.
Thus we can multiply this equation 
with vector fields and the result at a fixed point
depends only on the value of these vector fields there
but especially not on any derivatives. We deduce that
the same is true for both terms on the left-hand side. 

Taking any smooth non-vanishing vector field $\tilde\xi:\ol\Omega
\to\R^n$ we can project $\tilde\xi(x)$ to the tangential
hyperplane to $M(t)$ at $X=(x,-u(x,t))$ and normalize it such
that the result $\xi(x,t)$ satisfies $g_{ij}\xi^i\xi^j=1$.
We set $\xi_i=g_{ij}\xi^j$. In view of the above
considerations we get directly an evolution equation for
$h^j_i\xi_j\xi^i$. For simplicity we set $h^1_1:=
h^j_i\xi_j\xi^i$ and consider $h^1_1$ as a scalar
function. Here we wish to mention that the following
choice of $W$ has been proposed by the referee.
We get for $W:=\log h^1_1+\lambda\cdot v$,
where $\lambda\gg1$ is a constant to be fixed later,
the following evolution equation 
\begin{align}
\dot W-&\hat F^{ij}W_{ij}
=(-\lambda\cdot v+1)\cdot\hat F^{ij}h^k_ih_{kj}
+\frac{1}{h^1_1}\left(\hat F-\hat f-\hat F^{ij}h_{ij}
\right)h^k_1h^1_k\nonumber\\
&+\frac1{h^1_1}\left(\hat F^{kl,rs}h_{kl;1}h_{rs;}{}^1
+\frac1{h^1_1}\hat F^{ij}h^1_{1;i}h^1_{1;j}\right)
\label{h11+v evol}\\
&-\frac1{h^1_1}\hat f_{\alpha\beta}X^\alpha_1X^\beta_k
g^{k1}+\hat f_\alpha\nu^\alpha
+\lambda v^2\hat f_\beta X^\beta_iX^\alpha_j\eta_\alpha
g^{ij}-2\lambda\frac 1v\hat F^{ij}v_iv_j.\nonumber
\end{align}

For the $C^2$-a priori estimates at the boundary, we will
use a slightly different quantity, 
$$W:=\log\left(1+h^1_1\right)+\lambda\cdot v,$$
where $\lambda\gg1$ is again a constant to be fixed below.
Here we obtain the evolution equation
\begin{align}\label{mod W}
\dot W-&\hat F^{ij}W_{ij}=\left(-\lambda\cdot v+\frac{h^1_1}
{1+h^1_1}\right)\cdot\hat F^{ij}h^k_ih_{kj}
+\frac{h^k_1h^1_k}{1+h^1_1}\left(\hat F-\hat f
-\hat F^{ij}h_{ij}\right)\nonumber\\
&+\frac{1}{1+h^1_1}\left(\hat F^{kl,rs}h_{kl;1}h_{rs;}{}^1
+\frac{1}{1+h^1_1}\hat F^{ij}h^1_{1;i}h^1_{1;j}\right)\\
&-\frac1{1+h^1_1}\hat f_{\alpha\beta}X^\alpha_1X^\beta_k
g^{k1}+\frac{h^1_1}{1+h^1_1}\hat f_\alpha\nu^\alpha\nonumber\\
&+\lambda v^2\hat f_\beta X^\beta_iX^\alpha_j\eta_\alpha g^{ij}
-2\lambda\frac1v\hat F^{ij}v_iv_j.\nonumber
\end{align}

In the remaining part of this section we prove an estimate
for the terms in this evolution equation that contain 
derivatives of the second fundamental form.

\begin{lemma}\label{matr inequ}
Let $\left(a^{ij}\right)$ and $(A_{ij})$ be symmetric 
$n\times n$-matrices. Assume that $(A_{ij})$ is positive
semi-definite
and that $\left(a^{ij}\right)$ is positive definite with
inverse $(\tilde a_{ij})$. Then we have the inequality
$$-a^{ij}A_{ij}+\frac 1{\tilde a_{11}}A_{11}\le0.$$
\end{lemma}
\begin{proof}
For two positive semi-definite symmetric matrices $\left(
b^{ij}\right)_{1\le i,\, j\le n}$ and $\left(c_{ij}
\right)_{1\le i,\, j\le n}$ we can choose an orthogonal
basis transformation such that one of these matrices is 
diagonal and obtain that $b^{ij}c_{ij}\ge0$. This inequality 
is of course also valid in the original basis. 
So we have to prove that 
$$a^{ij}-\delta^i_1\delta^j_1\frac 1{\tilde a_{11}}=:d^{ij}$$
is positive semi-definite. We make a change of bases corresponding
to a block diagonal matrix of the form $\left(\begin{array}{cc} 
1&0\\0&T\\\end{array}\right)$, where $T$ is an orthogonal
$(n-1)\times(n-1)$-matrix and may thus assume that
$(d^{rs})_{2\le r,\,s\le n}$ and $(a^{rs})_{2\le 
r,\,s\le n}$ are diagonal. Note that the definition of
$\left(d^{ij}\right)$ is invariant under this special
transformation. For matrices of this form we wish to state
a useful equality
\begin{equation}\label{usef inequ}
\det\left(\begin{array}{ccccc}
a^{11} & a^{12} & \cdots  & \cdots & a^{1n} \\
a^{12} & a^{22} & 0       & \cdots & 0      \\
\vdots & 0      & \ddots  & \ddots & \vdots \\ 
\vdots & \vdots & \ddots  & \ddots & 0      \\ 
a^{1n} & 0      & \cdots  & 0      & a^{nn} \\
\end{array}\right)=\prod\limits_{i=1}^n a^{ii}
-\sum\limits_{i>1}\left|a^{1i}\right|^2\cdot 
\prod\limits_{\genfrac{}{}{0pt}{}{j\neq i}{j>1}} a^{jj}.
\end{equation}
We compute
$$\tilde a_{11}=\frac{\det\left(a^{rs}\right)_{2\le r,\, s\le n}}
{\det\left(a^{ij}\right)_{1\le i,\, j\le n}}
=\frac{\prod\limits_{i=2}^{n}a^{ii}}{\prod\limits_{i=1}^n
a^{ii}-\sum\limits_{i>1}\left|a^{1i}\right|^2\cdot
\prod\limits_{\genfrac{}{}{0pt}{}{j\neq i}{j>1}}a^{jj}},$$
so we get
$$d^{11}=a^{11}-\frac1{\tilde a_{11}}=
\frac{\sum\limits_{i>1}\left|a^{1i}\right|^2\cdot
\prod\limits_{\genfrac{}{}{0pt}{}{j\neq i}{j>1}}a^{jj}}
{\prod\limits_{i=2}^{n}a^{ii}}.$$
To prove that $\left(d^{ij}\right)$ is positive semi-definite,
we show that $\left(\tilde d^{ij}\right)$ is positive definite,
where 
$$\tilde d^{ij}=\left\{\begin{array}{cl}
d^{ij}, & i + j > 2,\\
d^{11}+\varepsilon, & i = j = 1,\end{array}\right.$$
with $\varepsilon>0$. Then $\varepsilon\downarrow 0$
gives the result. To see the positivity of $\left(\tilde d^{ij}
\right)$, we show that the
subdeterminants $\det\left(\tilde d^{ij}\right)_{k\le
i,\,j\le n}$ are positive for $1\le k\le n$. For $k>1$, this
is obvious. If $k=1$,
we use again  Formula (\ref{usef inequ}) and get 
$$\det \tilde d^{ij}
=\sum\limits_{i>1}\left|a^{1i}\right|^2\cdot
\prod\limits_{\genfrac{}{}{0pt}{}{j\neq i}{j>1}}a^{jj}+
\varepsilon\cdot\prod\limits_{i=2}^{n}a^{ii}-
\sum\limits_{i>1}\left|a^{1i}\right|^2\cdot
\prod\limits_{\genfrac{}{}{0pt}{}{j\neq i}{j>1}}a^{jj}>0,$$
so we obtain the claimed inequality.
\end{proof}

\begin{corollary}\label{3rd est cor}
Let $F\in\left(\tilde K\right)$ be a curvature function. Using
the notation introduced in this section, especially the
definition of $h^1_1$, we get
$$\hat F^{kl,rs}h_{kl;1}h_{rs;}{}^1+\frac1{h^1_1}
\hat F^{ij}h^1_{1;i}h^1_{1;j}\le0.$$
\end{corollary}
\begin{remark}
Note that we did neither assume that $h^1_1$ is the maximal
eigenvalue of $h^j_i$ nor that $h^j_i$ is diagonal. So this
corollary can be seen as a generalization of Lemma 1.3    
in \cite{cgjdg1996}.
\end{remark}
\begin{proof}
Note that $h_{rs;}{}^1=h_{rs;1}$, use the definition of the 
class $\left(\tilde K\right)$ and apply Lemma \ref{matr inequ}
with
$$\hat F^{kr}h_{kl;1}h_{rs;1}=A_{ls}\mbox{~~and~~}
\tilde h^{ls}=a^{ls}.$$ 
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$C^2$-estimates}\label{C2 estimates}
Making $T$ slightly smaller we may assume the existence
of a solution to our flow equation (\ref{flow eqn})
on the compact time interval $[0,T]$. This is no
restriction as the a priori estimates obtained will
not depend on $T$. 

\begin{lemma}\label{u beta}
For a solution of our flow equation (\ref{flow eqn}),
we have the following bounds for partial derivatives of $u$
on $\partial\Omega$,
$$u_{\tau\beta}=0\quad\mbox{and}\quad
|u_{\beta\beta}|\le(c(\varepsilon)+\varepsilon\cdot M)
\quad\mbox{for~any~}\varepsilon>0,$$
where $\tau$ denotes a vector tangential to $\partial
\Omega$, $\beta^k$ is an abbreviation for $h_{p_k}$,
and 
$$M:=\sup\limits_{\overline{\Omega}\times[0,T]}
|D^2u|.$$
\end{lemma}
\begin{proof}
This estimate is a parabolic version of the respective
estimate in \cite{ju Weingarten}.
We use indices to denote partial derivatives and 
differentiate the boundary condition $h(Du)=0$
on $\partial\Omega$ tangentially to obtain
$$u_{\tau\beta}=0\quad\mbox{on~}\partial\Omega.$$
To prove our second assertion, we apply the linear
operator $L$ defined by
$$L\tilde w:=-\dot{\tilde w}+v\hat F_{u_{ij}}\tilde w_{ij}
+v\hat F_{p_i}\tilde w_i+\left(\hat F-\hat f\right)
v_{p_i}\tilde w_i$$
to $w:=h(Du)$ and obtain using (\ref{dot uk}),
(\ref{rel hom}) or Lemma \ref{properties of F*...}
$$|Lw|\le(c(\varepsilon)+\varepsilon\cdot M)\cdot 
\tr\hat F_{u_{ij}}$$
for any $\varepsilon>0$. Applying a barrier function 
similar to that used near
Equation (\ref{estimate for theta}), multiplied
with $c(\varepsilon)+\varepsilon\cdot M$, we obtain the
claimed estimate for $u_{\beta\beta}$. 
\end{proof}

\begin{lemma}[Interior estimates]\label{int est}
For a solution of our flow equation (\ref{flow eqn}),
we can estimate the second derivatives of $u$ in
$\Omega\times[0,T]$ compared to those at the
parabolic boundary, more precisely
\begin{eqnarray*}
\sup\limits_{\Omega\times[0,T]}|D^2u|&\le&
c\cdot\left(1+\sup\limits_{(\partial\Omega\times[0,T])\cup
(\Omega\times\{0\})}|D^2u|\right)\\
&\le&c\cdot\left(1+\sup\limits_{\partial\Omega\times[0,T]}
|D^2u|\right).
\end{eqnarray*}
\end{lemma}
\begin{proof}
We may assume that for $\lambda\gg1$ to be fixed below,
$$(x,t,\xi)\mapsto\log\left(\frac{h_{ij}\xi^i\xi^j}
{g_{ij}\xi^i\xi^j}(x,t)\right)
+\lambda\cdot v,$$
where $(x,t)\in\ol\Omega\times[0,T]$ and $\xi\in\R^n\setminus
\{0\}$, attains its maximum in $(x_0,t_0,\xi_0)$ with
$(x_0,t_0)\in\Omega\times(0,T]$, i.\ e.\ $(x_0,t_0)$ 
does not belong to the parabolic boundary. Here we 
identified $(x,t)$ and $(\{x\}\times\R)\cap M(t)$.
We construct a vector field $\xi$ near $x_0$ as in 
Section \ref{geom evol eqns} and get
$$\left.\left(\log h^1_1+\lambda\cdot v\right)\right|_{(x_0,t_0)}
\ge\left.\left(\log h^1_1+\lambda\cdot v\right)\right|_{(x,t)},$$
especially for $x$ near $x_0$, $t\le t_0$. 
We may assume that $h^1_1(x_0,t_0)\ge1$. In the following
calculations in this proof, terms are always evaluated at
$(x_0,t_0)$. We use (\ref{h11+v evol}), Corollary \ref{3rd est cor}
(or the respective estimate of \cite{cgjdg1996})
and the parabolic maximum principle to obtain
\begin{equation}\label{log est}
0\le(-\lambda\cdot v + 1)\cdot\hat F^{ij}h^k_ih_{kj}+\frac1{h^1_1}
\left(\hat F-\hat f-\hat F^{ij}h_{ij}\right)h^k_1h^1_k
+c\cdot(1+\lambda),
\end{equation}
where we have already used our lower order estimates obtained
in Section \ref{oblique est} and the gradient bound following
from the boundary condition. As $\lambda\ge1$, $v\ge1$, we can 
apply (\ref{* inequ}) to get
$$0\le(-\lambda\cdot v+1)\cdot\varepsilon_0\cdot F\cdot H+
c\cdot H + c\cdot(1+\lambda).$$
Here $H=g^{ij}h_{ij}$ is the mean curvature of $M$.
We note that $F$ is uniformly bounded below by a positive
constant, so fixing $\lambda$ sufficiently large shows that
$H$ is bounded above. As $M$ is convex, we obtain immediately
the claimed inequality. 
\end{proof}

It remains to bound the second derivatives of $u$ on
$\partial\Omega\times(0,T]$. The next lemma, see also
\cite{ju Weingarten}, reduces this
estimate to an estimate for tangential directions.

\begin{lemma}\label{tang dom}
For a solution of our flow equation (\ref{flow eqn}) 
we have
$$\sup\limits_{\ol{\Omega}\times[0,T]}|D^2u|\le
c\cdot\left(1+\sup\limits_\tau u_{\tau\tau}\right),$$
where $\tau$ runs through all directions, i.\ e.\
vectors with $|\tau|=1$, tangential
to $\partial\Omega$.
\end{lemma}
\begin{proof}
We consider a fixed point in $\partial\Omega\times[0,T]$.
Let $\xi$ be any direction in $\R^n$. This direction 
can be decomposed as
$$\xi=a\tau+b\beta,$$
where $\tau$ is a tangential direction to $\partial\Omega$
at the fixed point and $\beta^k=h_{p_k}(Du)$ there. The 
constants $a,\,b\in\R$ are uniformly a priori bounded
due to our strict obliqueness estimates. Using Lemma~%
\ref{u beta} we get
$$u_{\xi\xi}=a^2u_{\tau\tau}+2abu_{\tau\beta}+b^2u_{\beta
\beta}
\le c\cdot\left(\sup\limits_\tau u_{\tau\tau}+c(\varepsilon)
+\varepsilon\cdot M\right)$$
with $M$ as in the cited lemma. Fixing $\varepsilon>0$
sufficiently small and using Lemma \ref{int est} 
gives the claimed estimate.
\end{proof}

In the next lemma we bound the second tangential 
derivatives of $u$ on $\partial\Omega\times[0,T]$.
We modify techniques of \cite{ju Weingarten}.
     
\begin{lemma}\label{cov est}
For a solution of our flow equation (\ref{flow eqn}),
the second derivatives of $u$ are a priori bounded,
$$\sup\limits_{\ol{\Omega}\times[0,T]}|D^2u|\le c.$$
\end{lemma}
\begin{proof}
We proceed similarly as in the proof of Lemma \ref{int est}
and use covariant derivatives.
We may assume that 
$$(x,t,\xi)\mapsto\log\left(1+\frac{h_{ij}\xi^i\xi^j}{g_{ij}\xi^i\xi^j}
(x,t)\right)+\lambda\cdot v(x,t),$$
where $(x,t)\in\partial\Omega\times[0,T]$ and $0\neq\xi$ runs 
through vectors tangentially to $\ol{M(t)}$ and
$\partial\Omega\times\R$, attains its maximum in $(x_0, t_0,
\xi_0)$ with $t_0>0$.
Here we identified $(x,t)$ and $(\{x\}\times\R)\cap
\ol{M(t)}$. Constructing a vector field $\xi$ -- that has to
be tangential on $\partial\Omega$ -- as in Section
\ref{geom evol eqns} near $x_0$, we get for 
$(x,t)\in\partial\Omega\times[0,T]$
\begin{equation}\label{h11 big}
\left(\log\left(1+h^1_1\right)+\lambda\cdot v\right)(x_0,t_0)\ge 
\left(\log\left(1+h^1_1\right)+\lambda\cdot v\right)(x,t).
\end{equation}
We may assume that this inequality holds also for 
$(x,t)\in\ol{\Omega}\times\{0\}$ 
as otherwise the estimate claimed
in this lemma is obvious. 
Let $W:=\log\left(1+h^1_1\right)+\lambda\cdot v$. 
We use the evolution equation (\ref{mod W}),
Corollary \ref{3rd est cor} and lower order estimates
to obtain
$$\dot W-\hat F^{ij}W_{ij}\le(-\lambda\cdot v+1)\cdot
\hat F^{ij}h^k_ih_{kj}+c\cdot\frac{h^k_1h^1_k}{1+h^1_1}
+c\cdot(1+\lambda).$$
As $\lambda\ge1$, $v\ge1$, we can use (\ref{* inequ}).
For the second term we use that, due to the convexity
of $u$, $(u_{1k})^2\le u_{11}\cdot u_{kk}$ and get
$$\dot W-\hat F^{ij}W_{ij}\le(-\lambda\cdot v+1)\cdot
\varepsilon_0\cdot F\cdot H+c\cdot(1+H+\lambda).$$
Fixing $\lambda$ sufficiently large, we see that
$$\dot W-\hat F^{ij}W_{ij}\le c.$$
We set $C_\delta:=B_\delta(x_0)\times\R$ and
$$Lw:=\dot w-\hat F^{ij}w_{ij}.$$
For small $\delta>0$ and
$$\tilde W:=\frac{W}{W(x_0,t_0)}-1,$$
we get
$$\left\{\begin{array}{rcll}
L\tilde W&\le&c&\mbox{on~}M\cap C_\delta,\\
\tilde W&\le&c&\mbox{on~}M\cap C_\delta,\\
\tilde W&\le&0&\mbox{on~}\partial M\cap C_\delta,\\
\tilde W&\le&0&\mbox{on~}M(0),\\
\tilde W&=&0&\mbox{at~}(x_0,t_0).
\end{array}\right.$$
We start to construct a barrier which will be used to obtain
the claimed estimate. The main part of this barrier function
consists of 
$$\vartheta:=-d+\mu d^2,$$
where $d=\dist(\cdot,\partial\Omega\times\R)$ is the Euclidean
distance to the cylinder over $\partial\Omega$ and $\mu\gg1$
will be fixed later-on. Direct computations yield on $M\cap
C_\delta$
\begin{eqnarray*}
L\vartheta&\le&\hat F^{ij}d_{\alpha\beta}X^\alpha_iX^\beta_j
-2\mu\hat F^{ij}d_\alpha X^\alpha_id_\beta X^\beta_j\\
&&{}+c\cdot(1+\mu\cdot\delta)
+c\cdot\mu\cdot\delta\cdot\tr\hat F^{ij},\quad
\tr\hat F^{ij}\equiv\hat F^{ij}g_{ij}.
\end{eqnarray*}
We use that in a Euclidean coordinate system $d_{\alpha\beta}$
is equal to the respective partial derivatives. The strict
convexity of $\partial\Omega$ gives for a positive constant
$\varepsilon>0$ that depends only on the principal
curvatures of $\partial\Omega$
\begin{eqnarray*}
L\vartheta&\le&-2\varepsilon\cdot\tr\hat F^{ij}-\mu\hat F^{ij}
d_\alpha X^\alpha_id_\beta X^\beta_j\\
&&{}+c\cdot(1+\mu\cdot\delta)+c\cdot\mu\cdot\delta\cdot
\tr\hat F^{ij}.
\end{eqnarray*}
By virtue of Lemma \ref{three equivalences} we can fix 
$\mu$ sufficiently large and then $\delta$ sufficiently
small to control the third term on the right-hand side. 
Fixing $\delta>0$ even smaller if necessary, we can absorb
the fourth term and get
$$L\vartheta\le-\varepsilon\cdot\tr\hat F^{ij}.$$
Further on we may assume that $\delta$ is so small that
$$\vartheta\le0\quad\mbox{on~}\partial C_\delta.$$
As a barrier function we choose
$$\Theta:=A\vartheta-B\cdot|x-x_0|^2+\tilde W,$$
where $|x-x_0|$ denotes the Euclidean distance for 
points in $\Omega$ and is evaluated on $M$ by projecting
$\Omega\times\R$ orthogonally to $\Omega$. We fix
$B\gg1$ to obtain an appropriate behavior on the boundary 
and then $A$ sufficiently large to obtain an appropriate
sign in the differential inequality, more precisely
$$\left\{\begin{array}{rcll}
L\Theta&\le&0&\mbox{on~}M\cap C_\delta,\\
\Theta&\le&0&\mbox{on~}\partial(M\cap C_\delta),\\
\Theta&\le&0&\mbox{on~}M(0),\\
\Theta&=&0&\mbox{at~}(x_0,t_0).
\end{array}\right.$$
Thus the maximum principle implies that $\Theta\le0$ in
$M\cap C_\delta$. We consider $\Theta$ as being defined
on $\ol{\Omega}\times[0,T]$, use partial derivatives
and get
$$\Theta_\beta\equiv h_{p_k}\Theta_k\le0
\quad\mbox{at~}(x_0,t_0).$$
Direct computations, (\ref{h11 big}), (\ref{h def}) and
Lemma \ref{tang dom} imply that
\begin{equation}\label{u11 beta}
u_{11\beta}\le c\cdot (c(\varepsilon)+\varepsilon\cdot u_{11})
\cdot(1+u_{11})\quad\mbox{at~}(x_0,t_0),
\end{equation}
where we have assumed that $\xi$ corresponds to the 
tangential direction $e_1$.
We differentiate the boundary condition twice and get at
$(x_0,t_0)$
$$h_{p_kp_l}u_{k1}u_{l1}+h_{p_k}u_{k11}+h_{p_k}u_{kn}
\omega_{11}=0,$$
where $\omega$ is a function such that locally
$\partial\Omega=\graph\left.\omega\right|_{\R^{n-1}}$
and $D\omega=0$ at the point corresponding to $x_0$. 
The index $n$ corresponds to a direction orthogonal
to $\partial\Omega$. Combining this with Inequality
(\ref{u11 beta}) and Lemma \ref{tang dom} as above, 
we get at $(x_0,t_0)$
$$-h_{p_kp_l}u_{k1}u_{l1}\le c\cdot\left(c(\varepsilon)+
\varepsilon\cdot(u_{11})^2\right).$$
As $h$ is strictly concave we can estimate the
left-hand side from below by 
$\inf\limits_k\,(-h_{p_kp_k})\cdot(u_{11})^2>0$, thus
fixing $\varepsilon>0$ sufficiently small bounds
$u_{11}$ and the claimed estimate follows.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of the main theorem}\label{proof main thm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Longtime existence}
Here and in the following we may restrict our considerations
to time intervals starting at $\varepsilon>0$ instead of
$0$. Thus we may ignore questions concerning compatibility
conditions and smoothness at $t=0$. We get uniform 
$C^2$-estimates for the partial derivatives of $u$ and a positive 
lower bound for $F$ and conclude that the flow operator  
is uniformly parabolic and concave. So we can apply the results  
of Chapter 14 in \cite{lieberman} to obtain uniform  
$C^{2,\alpha}$-estimates for $u$, with a small positive 
constant $\alpha$. Then standard Schauder  
estimates \cite{lady, lieberman} imply uniform bounds 
of the form $\left\Vert D^k u\right\Vert_{C^0\left(\ol\Omega\right)}
\le c_k$ for all $k\ge 1$. The estimate
for the function $u$ itself has been obtained in 
Corollary \ref{C0 est} and is not uniform in time. 
It follows from the considerations
concerning shorttime existence that a smooth solution of  
(\ref{flow eqn}) exists for all $t\ge 0$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Convergence to a translating solution}
We finish the proof of our Main Theorem \ref{main thm}
by showing that our solution that exists for all positive
times converges to a translation solution. In this section
we use partial derivatives. 

A similar proof can be found in \cite{awcvp1994}, where the existence
of a translating solution is established differently. In our
situation, however, the existence of a translating solution
is in general not obvious. 

We fix $t_0>0$ and establish
a boundary value problem fulfilled by
$$w(x,t):=u(x,t)-u(x,t+t_0).$$
By the mean value theorem we find a positive definite
matrix $\left(a^{ij}\right)$ and a vector field 
$\left(b^i\right)$ -- both
depending on $x$ and $t$ -- such that
$$\dot w=a^{ij}w_{ij}+b^iw_i\quad\mbox{in~}\ol{\Omega}
\times(0,\infty).$$
The boundary value condition for $w$ is derived as follows.
For any function $h:\R^n\to\R$ with $\{h=0\}=\partial\Omega^*$
and for any smooth strictly convex function $u:\ol{\Omega}\to\R$,
the boundary condition $Du(\Omega)=\Omega^*$ is equivalent
to $h(Du)=0$ on $\partial\Omega$. We have proved that 
$h_{p_k}(Du)\nu^k$ is uniformly bounded from below
by a positive constant on $\partial\Omega$, if $|\nabla h|=1$
on $\partial\Omega^*$ and $\nabla h$ points inside $\Omega^*$
there. Here we use $h=\min\{\dist(\cdot,
\partial\Omega^*),\,\varepsilon\}$ for $\varepsilon>0$ 
sufficiently small. We could also mollify $h$ slightly
near $\{\dist(\cdot,\partial\Omega^*)=\varepsilon\}$ to
obtain a smooth function $h$. We get
\begin{eqnarray*}0&=&h(Du(x,t))-h(Du(x,t+t_0))\\
&=&\int\limits_0^1h_{p_k}(\tau Du(x,t)+(1-\tau)Du(x,t+t_0))
\,d\tau\cdot w_k\equiv\beta^kw_k,
\end{eqnarray*}
so for $\varepsilon>0$ sufficiently small, $\beta^k$ is almost
equal to 
$$\sigma\cdot h_{p_k}(Du(x,t))+(1-\sigma)\cdot 
h_{p_k}(Du(x,t+t_0)),$$
for some $\sigma\in[0,1]$. Since we have uniform obliqueness
estimates during the evolution, it is possible to fix
$\varepsilon>0$ sufficiently small, depending only on the
obliqueness estimates and on the domain $\Omega^*$, such that
$\beta$ as defined above is a uniformly strictly oblique
vector field, i.\ e.\ $\beta^k\nu_k\ge\frac{1}{c}>0$. 
\par
The strong maximum principle implies that
$$\osc(w,t):=\osc(w(\cdot,t))=
\max\limits_{x\in\ol{\Omega}}w(x,t)-\min\limits_%
{x\in\ol{\Omega}}w(x,t)$$
is a strictly decreasing function in time or $w$ is constant.
Next, we will exclude the case when $\osc(w,t)$ is strictly
decreasing but tends to a positive constant $\varepsilon>0$ 
as $t\to\infty$. If $\osc(w,t)\to\varepsilon>0$, we
choose a sequence $t_n\to\infty$ and consider for
$(x,t)\in\ol\Omega\times[-t_n,\infty)$ and for fixed
$x_0\in\Omega$
\begin{equation}\label{tn sequences}
u(x,t+t_n)-u(x_0,t_n)\mbox{~~and~~}
u(x,t+t_0+t_n)-u(x_0,t_0+t_n).
\end{equation}
Our a priori estimates for the derivatives of $u$ yield for
$k\ge1$ uniform estimates of the form
$$\left\vert D^k(u(x,t+t_n)-u(x_0,t_n))\right\vert\le c_k.$$
We wish to prove locally, i.\ e.\ for $\vert t\vert < T$,
uniform bounds in any $C^k$-norm. It remains to bound the
absolute value of $u(x,t+t_n)-u(x_0,t_n)$. Therefore we use
the $\dot u$-bound, the $|Du|$-bound obtained from the boundary
condition, and the convexity of the domain $\Omega$ to estimate
for any $(x,t)\in\ol\Omega\times(-T,T)$ and any $t_n>T$
\begin{eqnarray*}
\lefteqn{|u(x,t+t_n)-u(x_0,t_n)|\le}\\
&\le&|u(x,t+t_n)-u(x,t_n)|+|u(x,t_n)-u(x_0,t_n)|\\
&\le& T\cdot\sup|\dot u|+\diam(\Omega)\cdot\sup|Du|.
\end{eqnarray*}
A similar argument applies to the second sequence. Thus we get
locally uniform bounds for any $C^k$-norm for both sequences
in (\ref{tn sequences}) and can extract a subsequence of
$t_n$ (again called $t_n$) such that the limits -- as
$t_n$ tends to infinity -- of both sequences in (\ref{tn
sequences}), $\tilde u^\infty$ and $\tilde u^{t_0,\infty}$,
satisfy our flow equation in $\ol\Omega\times\R$.
We define $\tilde w:=\tilde u^\infty-\tilde u^{t_0,\infty}$ 
and show that $\osc(\tilde w,t)=\varepsilon$ for any $t\in\R$.
To see this we fix $t\in\R$ and use especially the monotonicity
of the oscillation
\begin{eqnarray*}
\lefteqn{\osc(\tilde w,t)=\osc\left(\tilde u^\infty(x,t)-\tilde u^{t_0,
\infty}(x,t)\right)}\\
&=&\osc\lim\limits_{n\to\infty}\left(u(x,t+t_n)-u(x_0,t_n)
-(u(x,t+t_0+t_n)-u(x_0,t_0+t_n))\right)\\
%&=&\lim\limits_{n\to\infty}\osc\left(u(x,t+t_n)-u(x_0,t_n)
%-(u(x,t+t_0+t_n)-u(x_0,t_0+t_n))\right)\\
&=&\lim\limits_{n\to\infty}\osc\left(u(x,t+t_n)
-u(x,t+t_0+t_n)\right)\\
%&=&\lim\limits_{\tau\to\infty}\osc(u(x,\tau)-u(x,\tau+t_0))\\
&=&\lim\limits_{\tau\to\infty}\osc(w,\tau)\\
&=&\varepsilon.
\end{eqnarray*}
This, however, is impossible, 
as the strong maximum principle, applied to $\tilde w=
\tilde u^\infty-\tilde u^{t_0,\infty}$, 
shows that $\osc(\tilde w,t)>0$ is only possible if
$\osc(\tilde w,t)$ is strictly decreasing in $t$.
So $\osc(w,t)\to 0$ as $t\to\infty$ and we obtain that
\begin{equation}\label{v inf}
u(x,t)-u(x,t+t_0)\to-v^\infty\cdot t_0\quad\mbox{as~}
t\to\infty,
\end{equation}
uniformly in $x\in\ol\Omega$, where $v^\infty$ is a constant 
that does not depend on time as for $t>T$ the parabolic 
maximum principle implies that
\begin{eqnarray*}
\lefteqn{\inf\limits_{x\in\Omega}\left(u(x,T)-u(x,T+t_0)\right)
\le u(x,t)-u(x,t+t_0)}\\
&\le&\sup\limits_{x\in\Omega}\left(u(x,T)-u(x,T+t_0)\right).
\end{eqnarray*}
As we will see later-on, the
constant $v^\infty$ has been introduced such that it equals
the velocity of any translating solution. For an arbitrary
sequence $t_n\to\infty$, we consider
$$u(x,t+t_n)-u(x_0,t_n),\quad(x,t)\in\ol{\Omega}\times
[-t_n,\infty).$$
In view of our a priori estimates we may extract a not
relabeled subsequence $t_n\to\infty$ such that
\begin{equation}\label{to u0}
u(x,t+t_n)-u(x_0,t_n)\to u^0(x,t)
\end{equation}
locally uniformly in $\ol{\Omega}\times\R$ 
in any $C^k$-norm as $n\to\infty$. Equations 
(\ref{v inf}) and (\ref{to u0}) give
$$u^0(x,t+t_0)=u^0(x,t)+v^\infty\cdot t_0\quad\mbox{for~}
(x,t)\in\ol{\Omega}\times\R.$$
The function $u^0$ is again a solution to our flow equation.
We repeat the argument given above with $(u^0,t_1)$, $t_1>0$,
instead of $(u,t_0)$ and obtain a solution $u^1$ of our flow
equation satisfying
$$u^1(x,t+t_i)=u^1(x,t)+v^\infty_i\cdot t_i\quad
\mbox{for~}(x,t)\in
\ol{\Omega}\times\R,\quad i\in\{0,\,1\},$$
where $v^\infty_0=v^\infty$. We claim that $v^\infty_0=v^\infty_1$.
To see this, we note first that induction gives for $k\in\Z$
$$u^1(x,t+k\cdot t_i)=u^1(x,t)+v^\infty_i\cdot k\cdot t_i.$$
Now, we fix $(x,t)\in\Omega\times\R$ and find for any $T,\,\delta>0$ 
numbers $n_i\in\N$
such that $n_0\cdot t_0>T$ and $|n_0\cdot t_0-n_1\cdot t_1|<\delta$.
We obtain
\begin{eqnarray*}
\delta\cdot\sup|\dot u|&\ge&|u^1(x,t+n_0\cdot t_0)
-u^1(x,t+n_1\cdot t_1)|\\
&=&|u^1(x,t)+v^\infty_0\cdot n_0\cdot t_0
-u^1(x,t)-v^\infty_1\cdot n_1\cdot t_1|\\
&=&|v^\infty_0\cdot n_0\cdot t_0-v^\infty_1\cdot n_1\cdot t_1|\\
&\ge&|v^\infty_0\cdot n_0\cdot t_0-v^\infty_1\cdot n_0\cdot t_0|
-|v^\infty_1|\cdot|n_0\cdot t_0-n_1\cdot t_1|\\
&\ge&|v^\infty_0-v^\infty_1|\cdot T-|v^\infty_1|\cdot\delta.
\end{eqnarray*}
For $T$ sufficiently large, this is only possible if $v^\infty_0
=v^\infty_1$. So we obtain for $(x,t)\in\ol\Omega\times\R$,
$i\in\{0,1\}$, and $k\in\Z$
$$u^1(x,t+k\cdot t_i)=u^1(x,t)+v^\infty\cdot k\cdot t_i.$$
We can either choose $t_0$ and $t_1$ as incommensurable positive
numbers or we can repeat the argument for appropriate $t_l>0$,
$l\in\N$, and consider a diagonal sequence. In both cases we
obtain a smooth function $u^\infty:\ol\Omega\times\R\to\R$
that satisfies our flow equation and
$$u^\infty(x,t+\tau)=u^\infty(x,t)+v^\infty\cdot\tau$$
for $x\in\ol\Omega$ and $t,\,\tau\in\R$. Thus we have 
established the
existence of a translating solution to our flow equation.
We remark that this solution is -- up to additive constants -- the
only translating solution of our flow equation. This follows
from the strong maximum principle applied to the difference
of two translating solutions similar as at the beginning
of this section. 
\par
Finally, we show that $u$ converges to a translating 
solution. As above we get a linear parabolic differential
equation for $W:=u-u^\infty$,
$$\left\{\begin{array}{r@{~=~}ll}
\dot W&a^{ij}W_{ij}+b^iW_i&\mbox{in~}\ol{\Omega}\times
(0,\infty),\\[0.3ex]
0&\beta^kW_k&\mbox{on~}\partial\Omega\times[0,\infty),
\end{array}\right.$$
with a strictly oblique vector field $\beta$. Then
we get that the oscillation of $W$ tends to zero, thus
$u-u^\infty$ tends to a constant $c_\infty$ as $t\to
\infty$. We can use interpolation inequalities of
the form
$$\Vert Dw\Vert^2_{C^0}\le c(\Omega)\cdot\Vert w\Vert_{C^0}\cdot
\left(\left\Vert D^2w\right\Vert_{C^0}
+\left\Vert Dw\right\Vert_{C^0}\right)$$
for $w=W-c_{\infty}$ and its derivatives
and get smooth convergence of $u$ to a translating
solution. This finishes the proof that any solution of our
flow equation (\ref{flow eqn}) exists for all positive
times and tends eventually smoothly to a translating 
solution.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{appendix}
\section*{Appendix}
\thesection{\Alph{section}}
\setcounter{section}{1}
\subsection{Space-like hypersurfaces in Minkowski space}
A result similar to Theorem \ref{main thm} holds for strictly
space-like hypersurfaces in Minkowski space. To obtain
$C^2$-estimates, we have to consider a barrier function
that contains $\lambda\cdot e^{\tilde v}$
instead of $\lambda\cdot v$. As the boundary
condition ensures that the hypersurfaces remain strictly 
space-like during the evolution, the rest of the proof 
remains almost unchanged.

\subsection{Prescribed curvature}\label{pre F}
If we assume in contrast to the assumptions above, that
$f:\ol{\Omega}\times\R\to\R$ is smooth, positive and
satisfies $f_z<0$, we can prove as in \cite{osks2001},
that our flow converges to a hypersurface of prescribed
curvature, provided that either
$$\frac{f_z}{f}\le-c_f<0$$
or the first two compatibility conditions for $u_0$,
$$\dot u\ge 0\quad\mbox{for~}t=0,$$
and the growth condition $\log f(\cdot, z)\to\infty$
for $z\to-\infty$
are fulfilled. (The derivative of $f$ with respect to the
second argument is denoted by $f_z$ and $f$ is evaluated
at $X=(x,-u(x))$, $f=f(x,-u)$.)

The a priori estimates obtained in the sections above
guarantee that we get a solution for all positive 
times with estimates as above.
Convergence to a hypersurface of prescribed curvature
follows then similarly as in \cite{osks2001}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hessian flows}\label{hessian flows}
We consider the second boundary value problem for non-parametric
logarithmic Hessian flows
\begin{equation}\label{ivp}
\left\{\begin{array}{r@{~=~}ll}
\dot u &\hat F(D^2u)-\log g(x,Du)
&\mbox{in~}\Omega\times[0,T),\\[0.3ex] 
Du(\Omega)&\Omega^*,&
\end{array}\right.
\end{equation}
on a maximal time interval $[0,T)$, $T>0$, $u:\ol{\Omega}
\times[0,T)\to\R$. The Hessian function $F$ belongs to the 
class $\left(\tilde K^*\right)$ or equals $S_{n,k}$,
$1\le k\le n-1$. If $F=S_{n,k}$ for some $k$, then $g$
has to be independent of the gradient of $u$, i.\ e.\
$g=g(x)$. We assume that $\Omega,\,\Omega^*\subset
\R^n$, $n\ge2$, are strictly convex domains, $u_0:\ol{\Omega}\to
\R$ is a uniformly strictly convex function, 
$Du_0(\Omega)=\Omega^*$, and that
$g:\ol{\Omega}\times\ol{\Omega^*}\to\R$,
where $\Omega$, $\Omega^*$, $u_0$, and $g$ are smooth.
As initial condition for $u$ we take 
$$\left.u\right|_{t=0}=u_0.$$
\par
It is known \cite{osks2001} that this initial value problem 
has a smooth solution $u:\ol{\Omega}\times(0,\infty)$ and
$u,\,\dot u,\,Du,$ and $D^2u$ are continuous up to $t=0$. 
For $t\in[\varepsilon,\infty)$, $\varepsilon>0$, we have uniform
bounds for all $C^k$-norms besides for $|u|$ that may increase
as follows
$$\Vert u(\cdot,t)\Vert_{C^0}\le\Vert u(\cdot,0)\Vert_{C^0}
+t\cdot\Vert\dot u(\cdot,0)\Vert_{C^0}.$$
These estimates are not stated explicitly in \cite{osks2001}, but
follow immediately from the calculations there. For the longtime
behavior of solutions we have the following result. 

\begin{theorem}\label{trans thm}
Under the assumptions stated above, $u$ converges smoothly to 
a translating solution $u^\infty$ with velocity $v^\infty$, 
i.\ e.\ $u^\infty(x,t)=u^\infty(x,0)+v^\infty\cdot t$, of 
(\ref{ivp}) as $t\to\infty$. The translating
solution $u^\infty$ is independent 
-- up to additive constants -- of the choice of $u_0$. If 
$F(D^2u)=\det D^2u$ and $g(x,p)=\frac{g_1(x)}{g_2(p)}$
with smooth positive functions $g_1$ and $g_2$,
then $v^\infty$ is given by
$$v^\infty:=\log\int\limits_{\Omega^*}g_2(p)\,dp
-\log\int\limits_\Omega g_1(x)\,dx.$$
\end{theorem}
\begin{proof}
It follows from \cite{osks2001} that a solution to our 
initial value problem exists for all positive times.
Furthermore we get bounds for the $C^k$-norms as
in the proof of our Main Theorem \ref{main thm}
and thus longtime existence of solutions. As above we
conclude that our solutions converge to translating
solutions, that are unique up to additive constants. 

It remains to compute the velocity of 
a translating solution in the special case mentioned
above. Let $u$ be a translating solution. We get
$$v^\infty=\log\det D^2u-\log\frac{g_1(x)}{g_2(Du)}$$
or equivalently
$$g_1(x)\cdot e^{v^\infty}=\det D^2u\cdot g_2(Du).$$
We integrate this equation over $\Omega$ and get
$$e^{v^\infty}\cdot\int\limits_\Omega g_1(x)\,dx=
\int\limits_\Omega\det D^2u\cdot g_2(Du)\,dx
=\int\limits_{\Omega^*}g_2(p)\,dp,$$
where we have used the transformation rule.
This implies that $v^\infty$ is as claimed.
\end{proof}

\begin{remark}\label{exist rem}
In the special case of Theorem \ref{trans thm} 
when $g(x,p)=\frac{g_1(x)}{g_2(p)}$
and $F(D^2u)=\det D^2u$, 
it is possible to obtain the translating solution
directly as in Theorem~2~\cite{jucj1997}. 
The second boundary value problem
$$\left\{\begin{array}{r@{~=~}ll}
\det D^2u_\varepsilon & e^{\varepsilon u_\varepsilon+v^\infty}
\cdot\frac{g_1(x)}{g_2(Du)}&\mbox{in~}\Omega,\\[0.3ex]
Du_\varepsilon(\Omega)&\Omega^*,&
\end{array}\right.$$
is known \cite{jucj1997, osks2001} to have a solution 
$u_\varepsilon$ for $0<\varepsilon<1$. We integrate this
equation and use the transformation formula for integrals
$$e^{v^\infty}\cdot\int\limits_\Omega e^{\varepsilon u_\varepsilon}
\cdot g_1(x)\, dx=\int\limits_{\Omega^*}g_2(p)\,dp,$$
so we infer from the definition of $v^\infty$
$$\int\limits_\Omega e^{\varepsilon u_\varepsilon}\cdot
g_1(x)\,dx=\int\limits_\Omega g_1(x)\, dx,$$
and deduce that $u_\varepsilon$ is zero somewhere in $\Omega$.
Now, uniform $C^k$-a priori estimates follow from the
proofs in \cite{jucj1997, osks2001}. We let $\varepsilon\to0$,
extract a suitable subsequence of $u_\varepsilon$ and obtain
a solution $u$ of
\begin{equation}
\left\{\begin{array}{r@{~=~}ll}
v^\infty &\log\det D^2u
-\log g(x,Du)&\mbox{in~}\Omega,\\[0.3ex]
Du(\Omega)&\Omega^*.&\\[0.3ex]
\end{array}\right.
\end{equation}
Then we define 
$$u^\infty(x,t):=u(x)+v^\infty\cdot t,\quad(x,t)\in
\ol{\Omega}\times\R,$$
and get a solution $u^\infty$ of 
\begin{equation}
\left\{\begin{array}{r@{~=~}ll}
\dot u^\infty &\log\det D^2u^\infty
-\log g(x,Du^\infty)&\mbox{in~}\Omega\times\R,\\[0.3ex]
Du^\infty(\Omega)&\Omega^*&\\[0.3ex]
\end{array}\right.
\end{equation}
that moves by translation with velocity $v^\infty$.
\end{remark}
\end{appendix}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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