% O. Schnuerer, K. Smoczyk:
% Neumann and second boundary value problems for Hessian   
%   and Gau{\ss} curvature flows
% (Boundary Value Problems for Curvature Flows)
% Le probl\`eme \`a bord de Neumann et le deuxi\`eme 
%   probl\`eme \`a bord pour des flots hessiens et de 
%   courbure de Gau\ss{}
% May 2001.

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\begin{document}    
  
\title{Neumann and second boundary value problems for Hessian   
   and Gau{\ss} curvature flows}  
   
%    Information for first author   
\author{Oliver C.\ Schn\"urer}   
% Der Umlaut erzeugt overfull vbox!   
%    Address of record for the research reported here   
\address{Max Planck Institute for Mathematics in the Sciences,    
  Inselstr.\ 22-26, 04103 Leipzig, Germany}   
%    Current address   
%\curraddr{}   
\email{Oliver.Schnuerer@mis.mpg.de, Knut.Smoczyk@mis.mpg.de}   
%    \thanks will become a 1st page footnote.   
%\thanks{}   
   
%    Information for second author   
\author{Knut Smoczyk}   
%\address{Max Planck Institute for Mathematics in the Sciences,   
%  Inselstr.\ 22-26, 04103 Leipzig, Germany}   
%\email{Knut.Smoczyk@mis.mpg.de}   
%\thanks{}   
   
%    General info   
\renewcommand{\subjclassname}{%  
  \textup{2000} Mathematics Subject Classification}  
\subjclass{Primary 53C44; Secondary 35K20, 53C42}   
   
\date{May 2001.}   
   
%\dedicatory{}   
   
%\keywords{}   
   
\begin{abstract}   
We consider the flow of a strictly convex   
hypersurface driven by the  
Gau{\ss} curvature. For the Neumann boundary value  
problem and for the second boundary value problem we  
show that such a flow exists for all times and   
converges eventually to a solution of the prescribed  
Gau{\ss} curvature equation. We also discuss   
oblique boundary value problems and flows   
for Hessian equations.\\[2ex]
{\sc R\'esum\'e.}
Nous consid\'erons le flot d'une hypersurface strictement
convexe pilot\'e par la courbure de Gau{\ss}. Pour le probl\`eme 
\`a bord de Neumann et pour le deuxi\`eme probl\`eme \`a bord 
nous montrons qu'un tel flot existe pour tous les 
temps et converge ultimement vers une solution de l'\'equation de 
courbure de  Gau{\ss} prescrite. Nous discutons aussi des probl\`emes 
\`a condition de bord oblique et les flots pour des \'equations
hessiennes.
%Considerons l'\'evolution des hypersurfaces strictement convexes 
%pouss\'ees par la courbure de Gau{\ss}. Nous d\'emontrerons pour 
%la condition de Neumann et pour la condition deuxi\`eme \`a la  
%fronti\`ere, que notre equation admet une solution pour  
%toujours qui converge finalement vers une solution de
%l'\'equation \`a courbure de Gau{\ss} prescrite. Nous  
%\'evoquerons aussi des conditions oblique \`a la fronti\`ere 
%et des \'equations d'\'evolution pour des fonctions \`a hessien.    
\end{abstract}   
\maketitle   
   
\section{Introduction}   
\markboth{Oliver C.\ Schn\"urer and Knut Smoczyk}  
{Neumann and second boundary value problems for   
Hessian and Gau{\ss} curvature flows}  
This paper concerns -- in its first part --   
the deformation of convex graphs over bounded,  
convex domains  
$\Omega\subset \R{n}$, $n\ge 2$,   
with smooth boundary $\partial\Omega$  
to convex graphs with prescribed Gau\ss\ curvature  
and Neumann boundary condition. More precisely,  
let $u$ be a smooth strictly convex solution of   
\begin{equation}\label{flow equation}   
\left\{\begin{array}{rcll}  
\dot u&=&\Phi(\log\det(u_{ij})-\log f(x,u,Du))  
&\mbox{in~}\Omega\times[0,T),\\   
u_\nu&=&\varphi(x,u)&\mbox{on~}\partial\Omega\times[0,T),\\  
\left.u\right|_{t=0}&=&u_0&\mbox{in~}\Omega,  
\end{array}\right.  
\end{equation}  
for a maximal time interval   
$[0,T)$,  
where $f,\,\varphi:\overline{\Omega}\times\R{}\to \R{}$   
are smooth functions, $\nu$ denotes the inner unit normal  
to $\partial\Omega$  
and $u_0:\overline\Omega\to\R{}$, the initial value,   
is a smooth strictly convex  
function. Here $\Phi:\R{}\to\R{}$ is a smooth strictly   
increasing and concave function   
that vanishes at zero, i.e. $\Phi$ satisfies  
\begin{equation}  
\Phi(0)=0,\quad\Phi'>0,\quad\Phi''\le 0.\label{Phi 123}  
\end{equation}  
In the sequel we assume for simplicity $0\in\Omega$.  
  
To guarantee shorttime existence for (\ref{flow equation})   
and convergence to smooth graphs with prescribed Gau\ss\  
curvature we have to assume several structure conditions.  
These are  
\begin{eqnarray}  
\label{structure 1}  
\varphi_z&\equiv&\frac{\partial\varphi}{\partial z}\ge c_\varphi   
  >0,\\  
\label{structure 3}  
f&>&0\quad\mbox{and}\quad f_z\ge 0.  
\end{eqnarray}  
Moreover, we will always either assume  
\begin{equation}   
\label{structure 2}  
\frac{f_z}{f}\ge c_f>0  
\end{equation}   
or  
\begin{equation}  
\label{positive velocity}  
\Phi\left(\log\det (u_0)_{ij}-\log f(x,u_0,Du_0)\right)\ge 0.  
\end{equation}  
  
To guarantee smoothness up to $t=0$ it is necessary to assume  
the following compatibility conditions to be fulfilled on  
the boundary $\partial\Omega$ for any $m\ge0$  
\begin{equation}\label{compatibility condition}  
\left(\frac{d}{dt}\right)^m\left.\left(\nu^iu_i-\varphi(x,u)\right)  
\right|_{t=0}=0,  
\end{equation}  
where time derivatives of $u$, $u_i$, \ldots have to be substituted  
inductively by using $\dot u=\Phi$ and $\left.u\right|_{t=0}=u_0$.   
Applying Theorem 5.3, p.~320 \cite{lady} and the implicit   
function theorem, we obtain  
smooth shorttime existence up to $t=0$,   
see also \cite{cg2001}.  
  
During the flow, the smoothness of a solution guarantees that  
(\ref{compatibility condition}) is satisfied for any $m\ge 0$.  
So it is possible to extend a solution of the flow equation  
on a time interval $[0,T)$ to $[0,T]$ provided there are   
sufficient a priori estimates and then to $[0,T+\varepsilon)$  
 for a small $\varepsilon>0$. In this way we obtain existence  
 for all $t\ge0$ from the a priori estimates. The same  
procedure works also for the other boundary conditions   
considered in this paper.  
  
The main theorem for Neumann boundary conditions states  
\begin{theorem}\label{main 1}  
Assume that $\Omega$ is a bounded,   
strictly convex domain in $\R{n}$, $n\ge 2$,  
with smooth boundary.   
Let $f,\, \varphi:\overline{\Omega}\times\R{}\to\R{}$,   
be smooth functions that satisfy  
(\ref{structure 1})--(\ref{structure 3}).   
Let $u_0$ be a smooth, convex function that satisfies the compatibility  
conditions (\ref{compatibility condition}).   
Moreover, we assume that one  
of the conditions (\ref{structure 2}) or (\ref{positive velocity}) is  
fulfilled.  
Then a smooth solution  
of (\ref{flow equation}) exists for all $t\ge0$.  
As $t\to\infty$, the functions $\left.u\right\vert_t$ smoothly  
converge to a smooth limit function $u^\infty$ such that the graph  
of $u^\infty$ satisfies the Neumann boundary value problem  
\begin{equation}\label{stationary problem}  
\left\{\begin{array}{rcll}  
\det(u^\infty_{ij})&=&f(x,u^{\infty},Du^\infty)&\mbox{in~}\Omega,\\  
u^\infty_\nu(x)&=&\varphi(x,u^\infty)&\mbox{on~}\partial\Omega,  
\end{array}\right.  
\end{equation}  
where $\nu$ is the inward pointing unit normal of $\partial\Omega$.  
The rate of convergence is exponential provided (\ref{structure 2}) is  
satisfied.  
\end{theorem}  
  
When we assume condition (\ref{structure 2}), we obtain  
-- by using (\ref{compatibility condition}) only for $m=0$ -- a  
solution of (\ref{flow equation})   
which is smooth only for   
$t>0$ and the rate of convergence is exponential only in   
time intervals $[\varepsilon,\infty)$, $\varepsilon>0$.   
  
In the case when condition (\ref{positive velocity}) holds, we need   
only (\ref{compatibility condition}) for $m=0,\,1$ to obtain   
a solution of (\ref{flow equation}). Here $u$ approaches   
$u_0$ for $t\to 0$ only up to its fourth derivatives, where   
time derivatives have to be counted twice.   
  
In both cases, all the other claims of Theorem \ref{main 1}  
remain unchanged.  
  
\begin{remark}  
If we consider for a smooth function $\Psi:\R2\to\R{}$ the   
evolution equation  
\begin{equation}\nonumber  
\dot u=\Psi(\log\det u_{ij},\log f)  
\end{equation}  
and assume natural structure conditions, i.\ e.\   
concavity of $\Psi$, $\Psi_1>0$ and $\Psi(x,x)=0\quad\forall\,x$,  
then we prove in Lemma \ref{Phi gleich Psi} that   
there exists $\Phi:\R{}\to\R{}$ with $\Phi'>0$, $\Phi''\le 0$  
such that $\Psi$ has the following simpler form  
$$\Psi(x,y)=\Phi(x-y).$$  
\end{remark}  
  
\begin{example}  
 For $\Phi(x)=x$,   
our ansatz yields the logarithmic Gau{\ss}  
curvature flow  
\begin{equation*}  
\dot u=\log\det u_{ij}-\log g(x,u)   
-\mbox{$\frac{n+2}{2}$}\log\left(1+\vert Du\vert^2\right),  
\end{equation*}  
more precisely, the ''vertical`` velocity equals the difference  
of the logarithms of the actual and the prescribed  
Gau{\ss} curvature.   
Another interesting example is given by  
$\Phi(x)=1-e^{-\lambda x}$, $\lambda>0$, which gives the   
flow equation  
\begin{equation*}  
\dot u=1-\left(\frac{f(x,u,Du)}{\det u_{ij}}\right)^\lambda.  
\end{equation*}  
\end{example}  
  
In a second part, we consider the second boundary value problem  
for Hessian flow equations, more precisely, we solve the  
initial value problem  
\begin{equation}\label{flow equation second}  
\left\{\begin{array}{rcll}  
\dot u&=&\log F(D^2u)-\log g(x,u,Du)&\mbox{in~}\Omega  
\times[0,T),\\[0.3ex]  
Du(\Omega)&=&\Omega^*,&\\[0.3ex]  
\left.u\right|_{t=0}&=&u_0&\mbox{in~}\Omega,  
\end{array}\right.\end{equation}  
on a maximal time interval $[0,T)$, $T>0$. We assume that $\Omega,\,  
\Omega^*\subset\R{}^n$, $n\ge2$, are smooth strictly convex domains,  
$u_0:\overline{\Omega}\to\R{}$ is a smooth strictly convex  
function, $Du_0(\Omega)=\Omega^*$ (= $0$-th compatibility  
condition), $g:\overline{\Omega}\times\R{}\times
\overline{\Omega^*}\to\R{}$  
is a smooth positive function such that $g_z>0$. $F$ is a  
Hessian function of the class $\left(\tilde K^\star\right)$,  
for a precise definition we refer to Definition  
\ref{DefinitionClassK}. Here we remark only, that  
the class of Hessian functions considered includes   
especially $F(D^2u)=\det D^2u$. We will  
show that a smooth strictly convex solution of (\ref{flow  
equation second}) exists for all times, i.\ e.\ $T=\infty$, and  
converges  
smoothly to a solution $u^\infty$ of the elliptic second  
boundary value problem  
\begin{equation}\label{stationary problem second}  
\left\{\begin{array}{rcll}  
F(D^2u^\infty)&=&  
g(x,u^\infty,Du^\infty)&\mbox{in~}\Omega,\\[0.3ex]  
Du^\infty(\Omega)&=&\Omega^*,&  
\end{array}\right.  
\end{equation}  
when some structure conditions are fulfilled. The  
asymptotic behavior of $g$ is given by  
\begin{equation}\begin{array}{rcl@{\mbox{~as~}}l}  
\label{g asymptotics}  
g(x,z,p)&\to&\infty&z\to\infty,\\[0.2ex]  
g(x,z,p)&\to&0&z\to-\infty,  
\end{array}\end{equation}  
uniformly for $(x,p)\in\overline{\Omega}\times  
\overline{\Omega^*}$.  
 Furthermore we will always assume that there holds either  
\begin{equation}\label{g condition}  
\frac{g_z}{g}\ge c_g>0  
\end{equation}  
or  
\begin{equation}\label{dot u positive}\left\{\begin{array}{rcll}  
0&\le&F(D^2u_0)-\log g(x,u_0,Du_0)&\mbox{in~}\Omega,\\[0.3ex]  
&&\mbox{1st compatibility condition}&\mbox{on~}  
\partial\Omega,  
\end{array}\right.\end{equation}  
where the inequality means that $u_0$ is a subsolution.  
We remark that the  
boundary condition $Du(\Omega)=\Omega^*$ is equivalent  
to $h(Du)=0$ on $\partial\Omega$ for smooth strictly  
convex functions $u$, where $h:\R{}^n\to\R{}$ is a smooth   
strictly concave function such that $\left.h\right|_{\partial  
\Omega^*}=0$ and $|\nabla h|=1$ on $\partial\Omega^*$.  
 For the second boundary value problem the compatibility  
conditions read as follows  
\begin{equation}\label{m-th compatibility condition}  
\left(\frac{d}{dt}\right)^m\left.h(Du)  
\right|_{t=0}=0,\quad m\in\N,  
\end{equation}  
where derivatives of $u$ have to be replaced as above.  
  
 For the second boundary value problem,   
we obtain the following main theorem.  
\begin{theorem}\label{main theorem}  
Assume that $\Omega$, $\Omega^*$, $g$, $u_0$ and $F$ are  
as assumed above and either (i) (\ref{g condition}) or  
(ii) (\ref{dot u positive})  are satisfied. Then there  
exists a smooth strictly convex function $u:\overline{\Omega}  
\times(0,\infty)\to\R{}$ of (\ref{flow equation second}), i.\ e.\  
$T=\infty$, and $u$ converges smoothly to a solution $u^\infty$ of  
(\ref{stationary problem second}) as $t\to\infty$. Furthermore,  
$u$ is continuous up to its (i) second/(ii) fourth  
derivatives at $t=0$, where time derivatives have to  
be counted twice, and (i) gives exponential convergence  
$u\to u^\infty$ for $t\in[\varepsilon,\infty)$, $\varepsilon>0$.  
If (\ref{m-th compatibility condition}) is fulfilled for all  
$m\in\N$, then $u$ is smooth in $[0,T)$ and (i) gives  
exponential convergence to $u^\infty$ in $[0,\infty)$.  
\end{theorem}  
  
This result extends to Hessian quotient equations as follows.  
  
\begin{theorem}\label{Snk theorem}  
Theorem \ref{main theorem} holds also for $F=S_{n,k}$,  
$1\le k\le n-1$,  
when $g$ happens to be independent of the gradient of $u$,  
where $S_{n,k}(D^2u)$ is the quotient of the $n$-th and  
the $k$-th elementary symmetric polynomial of the  
eigenvalues of $D^2u$.  
\end{theorem}  
  
\begin{notation}  
Indices denote partial derivatives or vector components  
and are lifted and lowered   
with respect to $\delta_{ij}$ except for     
$\left(u^{ij}\right)$ that denotes the inverse of $(u_{ij})$.  
Indices $z$ and $p_i$ denote partial derivatives with   
respect to the argument used for the function $u$ and   
for its gradient, respectively, dots refer to time  
derivatives. We use the Einstein summation   
convention and sum over repeated Latin indices from $1$ to $n$.  
 For a vector $\nu$ we use $u_\nu\equiv u_i\nu^i$ with obvious  
generalizations to other quantities.  
We use $c$ to denote a positive and already estimated constant.  
Its value may change from line to line if necessary. We point  
out that the inequalities remain valid when $c$ is enlarged.  
A function $u:\Omega\times[0,T)\to\R{}$ is called (strictly) convex,  
if $u(\cdot, t)$ is (strictly) convex for every time $t\in[0,T)$.
A function $u:\Omega\to\R{}$ is called strictly convex, if the
eigenvalues of its Hessian are positive. This definition extends
to hypersurfaces and sets by using their principal curvatures.  
 Finally, we use $$\hat f=\log f$$ 
to denote the logarithm of a function
$f$.  
\end{notation}  
  
We briefly discuss the relation of our result with the existing  
literature.  
In \cite{Firey} smooth, compact, strictly convex and rotationally  
symmetric hypersurfaces in $\R3$  have been deformed by its Gau\ss\  
curvature to round points. The Gau\ss\ curvature flow  
\begin{equation}\nonumber  
\frac{d}{dt}F=-K\nu  
\end{equation}  
for smooth embeddings $F$ of hyperspheres in $\R{n+1}$ has been the  
subject in \cite{Andrews}. For the $n$-th root of $K$ this flow  
has been considered in \cite{Chow}. In \cite{chou wang, cgiumj2000}  
the authors use flow equations to prove existence theorems  
for closed hypersurfaces of prescribed curvature.  
 For Gau{\ss} curvature flows strict 
convexity is an essential assumption   
because then the flow becomes strictly parabolic. In addition  
the degenerate Gau\ss\ curvature flow with flat sides has been  
investigated  
in \cite{Daskalopoulos Hamilton}.   
There are also several papers about curvature flows with Dirichlet  
boundary condition, we only mention \cite{ivochkina}.  
The elliptic version of our flow equations  
(\ref{flow equation}), (\ref{flow equation second})  
has been explored in \cite{ltu,  
Urbas Hessian, Urbas Calc 1998, Urbas  
Crelle} by using the continuity method, see also \cite{Urbas  
Weingarten} for a related problem.  
Some of the techniques used there will be applied in our paper  
as well.   
  
The organization of our paper is as follows:   
In the first part, we study flow equations subject  
to prescribed Neumann boundary values. In section  
\ref{section 2} we prove uniform estimates for $\vert\dot u\vert$.   
This will be used in section \ref{section C0} to derive  
$C^0$-estimates.  
$C^1$-estimates then follow from \cite{ltu}. As a consequence we will   
obtain a uniform positive lower bound for $\det u_{ij}$.  
In section \ref{section 3} we derive $C^2$-estimates and in  
section \ref{section 4} we mention how to obtain  
H\"older regularity for the second derivatives of $u$  
and prove Theorem \ref{main 1}.   
In a second part we consider the second boundary value  
problem. In section \ref{Curvature functions and the   
Legendre transformation section} we introduce Hessian functions  
and a dual problem, next, we prove the strict obliqueness  
of our boundary condition. After the estimates for  
$\dot u$ and $u$ in section \ref{C0-estimates section},  
we give a quantitative version of our obliqueness result.  
In section \ref{C2-estimates section} we establish  
$C^2$-estimates and in section \ref{Proof of the  
main theorem section} we prove Theorem \ref{main theorem}.  
As far as the second boundary value problem is concerned,  
we will use methods of \cite{Urbas Hessian,   
Urbas Crelle} without mentioning this explicitly there.  
In the appendix we state  
generalizations to oblique boundary value problems for  
Hessian equations and indicate how to obtain the result for  
Hessian quotient equations.  
We remark that our results are parabolic versions   
of \cite{ltu, Urbas Hessian, Urbas Calc 1998}, so our results  
can be considered as alternative existence proofs using  
parabolic methods.  
  
This paper has been finished as the second author visited   
the program ``Nonlinear Partial Differential Equations'' of the  
Isaac Newton Institute for Mathematical Sciences, Cambridge,  
in 2001. He wants to thank the organizers for their invitation to  
this wonderful and very stimulating location.  
Both authors thank Claus Gerhardt for useful suggestions and  
discussions.  
In addition, the authors are indebted to J\"urgen Jost for his support  
at the Max Planck Institute for Mathematics in the Sciences where part  
of this paper has been written.   
  
%%%%%%   
\section{$\dot u$-estimates}\label{section 2}   
  
 For a constant $\lambda$ we define the function  
\begin{equation}\nonumber  
r:=e^{\lambda t}(\dot u)^2.  
\end{equation}  
An easy computation shows that (\ref{flow equation}) implies  
the following evolution equation for $r$  
\begin{equation}\label{evol r}  
\dot r=\Phi'u^{ij}r_{ij}-2e^{\lambda t}\Phi'u^{ij}\dot u_i\dot u_j  
-\Phi'\frac{f_{p_i}}{f}r_i+\left(\lambda  
-2\Phi'\frac{f_z}{f}\right)r.  
\end{equation}  
  
\begin{lemma}\label{bound for dot u}  
As long as a smooth convex solution of   
(\ref{flow equation}) exists we obtain the estimate  
\begin{equation}\nonumber  
\min\left\{\min_{t=0}\dot u,0\right\}\le\dot u\le  
\max\left\{\max_{t=0}\dot u,0\right\}.  
\end{equation}  
\end{lemma}  
\begin{proof}  
If $(\dot u)^2$ admits a positive local maximum  
in $x\in\partial\Omega$ for a positive time, 
then we differentiate the Neumann  
boundary condition and obtain from (\ref{structure 1})  
\begin{equation*}  
\left((\dot u)^2\right)_\nu=2(\dot u)^2\varphi_z>0  
\end{equation*}  
which contradicts the maximality of $(\dot u)^2$ at $x$.  
Now we choose $\lambda=0$ in (\ref{evol r}) and get   
\begin{equation}\nonumber  
\frac{d}{dt}(\dot u)^2\le\Phi'u^{ij}\left((\dot u)^2\right)_{ij}  
-\Phi'\frac{f_{p_i}}{f}\left((\dot u)^2\right)_i  
-2\Phi'\frac{f_z}{f}(\dot u)^2.  
\end{equation}  
So we obtain from   
(\ref{Phi 123}) and (\ref{structure 3}) that  
a positive increasing local maximum of $(\dot u)^2$ on  
$\overline\Omega\times[0,t_0]$   
cannot occur at an interior point of $\Omega$  
for any time $0<t_0<T$.   
\end{proof}  
  
\begin{corollary}\label{argument of Phi bounded}  
As long as a smooth convex solution of   
(\ref{flow equation}) exists we  
get a positive lower bound for $\Phi'$,  
$\frac{1}{c_\Phi}>\Phi'>c_{\Phi}>0$.  
\end{corollary}  
\begin{proof}  
This follows immediately as Lemma \ref{bound for dot u},  
$\Phi(0)=0$ and the strict monotonicity of $\Phi$  
give a bound for the argument of $\Phi$.  
\end{proof}  
  
\begin{lemma}\label{lower bound for u-dot}  
\label{dot u bounded}  
As long as a smooth convex solution of   
(\ref{flow equation}) exists we obtain the estimate  
\begin{equation}\nonumber  
\min\left\{\min_{t=0}\dot u,0\right\}\le\dot u e^{\lambda t}\le  
\max\left\{\max_{t=0}\dot u,0\right\}  
\end{equation}  
for $\lambda\le c_fc_\Phi$ provided (\ref{structure 2}) is fulfilled.  
\end{lemma}  
\begin{proof}  
This statement follows from Corollary \ref{argument of Phi bounded}  
and a proof similar to the  
proof of Lemma \ref{bound for dot u}.  
\end{proof}  
  
\begin{lemma}\label{u-dot remains positive}   
A solution of our flow (\ref{flow equation})  
satisfies $\dot u>0$ or    
equivalently $\Phi>0$ for $t>0$   
if $0\not\equiv\dot u\ge0$ for $t=0$.   
\end{lemma}   
\begin{proof}   
Differentiating the flow equation yields    
\begin{equation}\label{u-dot equation}   
\ddot u=\Phi'u^{ij}\dot u_{ij}-\Phi'\left(\hat f_z\dot u  
+\hat f_{p_i}\dot u_i\right),  
\end{equation}   
thus  
\begin{equation}\label{Gl 123}  
\frac{d}{dt}\left(\dot ue^{\lambda t}\right)  
=\Phi'u^{ij}\left(\dot ue^{\lambda t}\right)_{ij}  
-\Phi'\left(\hat f_z\dot ue^{\lambda t}  
+\hat f_{p_i}\left(\dot ue^{\lambda t}\right)_{i}\right)  
+\lambda\dot ue^{\lambda t}.  
\end{equation}    
We fix $t_0>0$ and a constant $\lambda>0$ such that  
$\lambda>\Phi'\hat f_z$ for $(x,t)\in\overline\Omega\times[0,t_0]$.  
 From (\ref{Gl 123}) and the strong parabolic maximum  
principle we see that $\dot ue^{\lambda t}$ has to vanish  
identically if it vanishes in $(x_0,t)\in\Omega\times(0,t_0)$,   
contradicting $\dot u\not\equiv 0$ for   
$t=0$. If $\dot ue^{\lambda t}=0$ for $x_0\in\partial\Omega$   
the Neumann boundary condition implies  
\begin{equation}   
\left(\dot ue^{\lambda t}\right)_\nu=\varphi_z  
\left(\dot ue^{\lambda t}\right)=0,\nonumber   
\end{equation}   
but this is impossible in view of the Hopf lemma applied to    
(\ref{Gl 123}) because $\lambda>\Phi'\hat f_z$.   
\end{proof}   
\begin{remark}  
The constant $\lambda$ in the previous proof depends on $t_0$. It can  
be chosen independent of $t_0$, if   
$\Phi' \hat f_z$ is uniformly bounded above  
and this is true, if $u$ is bounded in $C^1$.  
\end{remark}  
  
\section{$C^0$- and $C^1$-estimates}\label{section C0}  
\begin{remark}  
The strict convexity of $u$ and the fact that $\varphi(\cdot, z)\to  
\infty$ uniformly as $z\to\infty$ imply that $u$ is uniformly a  
priori bounded from above as $u_\nu=\varphi(x,u)$ on $\partial  
\Omega$.  
\end{remark}  
  
\begin{lemma}\label{lower estimates}  
Under the assumptions of Lemma \ref{lower bound for u-dot} we have the   
following lower bound for $u$  
\begin{equation}\nonumber  
u\ge\min_{t=0}u+\frac{1}{\lambda}\min\left\{\min_{t=0}\dot u,   
0\right\}  
\end{equation}  
for all $0<\lambda\le c_\Phi c_f$.  
\end{lemma}  
\begin{proof}  
This easily follows from Lemma \ref{lower bound for u-dot}  
\begin{eqnarray}  
u(x,t)&=&u(x,0)+\int\limits_0^t\dot u(x,\tau)d\tau\nonumber\\  
&\ge&u(x,0)+\min\left\{\min_{t=0}\dot u,   
0\right\}\int\limits_0^te^{-\lambda\tau}d\tau  
\nonumber\\  
&\ge&\min_{t=0}u+\frac{1}{\lambda}\min\left\{\min_{t=0}\dot u,   
0\right\}\nonumber.  
\end{eqnarray}  
\end{proof}  
  
\begin{lemma}[$C^1$-estimates]  
 For a smooth and convex solution $u$ of the flow equation  
(\ref{flow equation}),  
the gradient of $u$ remains bounded during the evolution.  
\end{lemma}  
\begin{proof}  
This follows from the $C^0$-estimates obtained so   
far and Theorem~2.2 in \cite{ltu}.  
\end{proof}  
  
\begin{remark}  
As long as a smooth solution $u$ of our flow equation   
(\ref{flow equation}) exists and  
$\log\det u_{ij}$ remains bounded, $u$ remains strictly convex  
provided $u_0$ is strictly convex.  
The quantity $\log\det u_{ij}$, however, stays bounded as both  
the argument of $\Phi$  (see Lemma \ref{bound for dot u},   
Corollary \ref{argument of Phi bounded})  
and $\log f$ are estimated. Finally, $\log f$ remains bounded  
as $|u|_1$ is a priori bounded.  
\end{remark}  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5  
  
\section{$C^2$-estimates}\label{section 3}   
  
\subsection{Preliminary results}   
   
We use $\nu$ for the inner unit normal of $\partial\Omega$   
and $\tau$ for a direction tangential to $\partial\Omega$.   
   
\begin{lemma}[Mixed $C^2$-estimates at the boundary]  
\label{mixed lemma}  
Let $u$ be a solution of our flow equation (\ref{flow equation}).   
Then the absolute value of $u_{\tau\nu}$ remains a priori bounded   
on $\partial\Omega$ during the evolution.   
\end{lemma}   
\begin{proof}   
We represent $\partial\Omega$ locally as $\graph\omega$ over  
its tangent plane at a fixed point $x_0\in\partial\Omega$   
such that locally  
$\Omega=\{(x^n,\hat x):x^n>\omega(\hat x)\}$.   
We differentiate the Neumann boundary condition    
\begin{equation}    
\nu^i(\hat x)u_i(\hat x,\omega(\hat x))=   
\varphi((\hat x,\omega(\hat x)),u(\hat x,\omega(\hat x))),   
\quad\hat x\in\R{n-1},\nonumber  
\end{equation}   
with respect to $\hat x^j$, $1\le j\le n-1$,   
\begin{equation}   
\nu^i_ju_i+\nu^iu_{ij}+\nu^iu_{in}\omega_j=   
\varphi_j+\varphi_n\omega_j+\varphi_zu_j+   
\varphi_zu_n\omega_j\nonumber   
\end{equation}   
and obtain at $x_0\equiv(\hat x_0,\omega(\hat x_0))\in\partial  
\Omega$ a bound for   
$\nu^iu_{ij}$ in view of the $C^1$-estimates and   
$D\omega(\hat x_0)=0$.   
Multiplying with $\tau^j$ gives the result.  
We remark that it is only possible to multiply the equation  
with a tangential vector as the differentiation with respect  
to $\hat x^j$ and so also $j$ correspond to tangential  
directions.  
\end{proof}   
   
\begin{lemma}[Double normal $C^2$-estimates at the boundary]  
\label{Double normal lemma}   
 For any solution of the flow equation (\ref{flow equation})    
the absolute value of  
$u_{\nu\nu}$ is a priori bounded from above on $\partial\Omega$.   
($u_{\nu\nu}>0$ also follows from the strict convexity of    
a solution.)   
\end{lemma}   
\begin{proof}   
We use methods known from the Dirichlet problem   
\cite{Minkowski space},   
where more details can be found and assume the same geometric  
situation as in the proof of Lemma \ref{mixed lemma}  
with $x_0\in\partial\Omega$.   
 From (\ref{flow equation}) we obtain   
\begin{equation}\nonumber   
\dot u_k=\Phi'u^{ij}u_{ijk}-\Phi'(\hat f_k+\hat f_zu_k+  
\hat f_{p_i}u_{ki})  
\end{equation}   
and define therefore   
\begin{equation}\nonumber   
Lw:=\dot w-\Phi'u^{ij}w_{ij}+\Phi'\hat f_{p_i}w_i,   
\end{equation}   
where we evaluate the terms by using the function $u$.  
 From the definition of $L$ it is easy to see that for    
appropriate extensions of $\nu$ and $\varphi$   
\begin{equation}\nonumber  
\left|L\left(\nu^ku_k-\varphi(x,u)\right)\right|\le c\cdot\trup,   
\end{equation}   
where - here and in the following - $c$ is an a priori bounded   
positive constant that may change its value as necessary.   
We define $\Omega_\delta:=\Omega\cap B_\delta(x_0)$ for    
$\delta>0$ sufficiently small and  
set   
\begin{equation}\nonumber   
\vartheta:=d-\mu d^2   
\end{equation}   
for $\mu\gg1$ sufficiently large where $d$ denotes the distance   
from $\partial\Omega$. We will show that $L\vartheta\ge   
\frac{\varepsilon}{3}\Phi'\tru$ for a small constant $\varepsilon>0$   
(depending only on a positive lower bound for the principal  
curvatures of $\partial\Omega$)  
in $\Omega_\delta$.   
\begin{eqnarray}\nonumber   
L\vartheta&=&-\Phi'u^{ij}d_{ij}+2\mu\Phi' u^{ij}d_id_j   
  +2\mu\Phi' u^{ij}dd_{ij}\nonumber\\  
&&{}+\Phi'\hat f_{p_i}(d_i-2\mu dd_i)\nonumber\\   
&\ge&-\Phi'u^{ij}d_{ij}+2\mu\Phi' u^{ij}d_id_j   
  -c\mu d\trup-c.\nonumber   
\end{eqnarray}   
We use the strict convexity of $\partial\Omega$, $d_i\approx   
\delta_{in}$, $|u^{kl}|\le\tru$, $1\le k,\, l\le n$, and the   
inequality for arithmetic and geometric means   
\begin{eqnarray}\label{L theta}   
L\vartheta&\ge&\varepsilon\,\Phi'\tru+\Phi'\mu u^{nn}   
  -c\mu\delta\trup-c\nonumber\\   
&\ge&\Phi'\frac{n}{3}\left(\det u^{ij}\right)^\frac{1}{n}   
  \cdot\varepsilon^\frac{n-1}{n}\cdot\mu^\frac{1}{n}   
  +\frac{2}{3}\varepsilon\,\Phi'\,\tru\\  
&&{}-c\mu\delta\trup-c.\nonumber   
\end{eqnarray}   
As $\det u^{ij}$ is a priori bounded from below by  
a positive constant in view of    
\begin{equation}\nonumber   
\det u^{ij}=(\det u_{ij})^{-1}   
  =\exp\left(-\hat f-\Phi^{-1}(\dot u)\right),  
\end{equation}   
we may choose $\mu$ so large that the first term in   
(\ref{L theta}) is greater than $c+1$. For $\delta\le   
\frac{1}{c\mu}\min\left\{1,\frac{1}{3}\varepsilon\right\}$   
we get   
\begin{equation}\nonumber   
L\vartheta\ge\frac{1}{3}\varepsilon\,\Phi'\,\tru   
\end{equation}   
and furthermore $\vartheta\ge 0$ on $\partial\Omega_\delta$   
if we choose $\delta$ smaller if necessary.   
\par   
 For constants $A,\,B>0$ consider the function   
\begin{equation}\nonumber   
\Theta:=A\vartheta+B|x-x_0|^2\pm(\nu^iu_i-\varphi(x,u))+l,   
\end{equation}   
where $l$ is an affine linear function such that 
$\Theta\ge0$ for $t=0$ and $l(x_0)=0$.
We fix $B\gg 1$, get $\Theta\ge0$ on $\partial\Omega_\delta$,   
and deduce for $A\gg B$ that $L\Theta\ge0$ as $\tru$ is    
bounded from below by a positive constant.   
The maximum principle yields $\Theta\ge0$   
in $\Omega_\delta$. As $\Theta(x_0)=0$ we have $\Theta_\nu(x_0)\ge0$   
which in turn gives immediately $|u_{\nu\nu}|\le c$.  
\end{proof}   
   
\begin{remark}\label{lambda remark}   
 From Section \ref{section C0} and the uniform estimates for $\dot u$  
we get for a fixed positive constant $\mu_0$   
\begin{equation}\nonumber   
\min\left\{\det u_{ij}, f\right\}\ge\mu_0>0.   
\end{equation}   
According to \cite{ltu} we obtain unique convex solutions    
$\psi_\rho\in C^2\left(\overline{\Omega}\right)$ for   
$0\le\rho\le 1$ of the boundary value problem   
\begin{equation}\nonumber   
\left\{\begin{array}{rcll}   
\det\psi_{ij}&=&\frac{1}{2}\mu_0&\mbox{in~}\Omega,\\   
\psi_\nu&=&\varphi(x,\psi+\rho|x|^2)-2\rho\langle x,\nu\rangle&   
\mbox{on~}\partial\Omega   
\end{array}\right.   
\end{equation}   
such that $|\psi_\rho|_{2,\overline{\Omega}}\le c$ and   
$\psi_{ij}\ge\lambda\delta_{ij}$ for positive constants  
independent of $\rho$. Fix $\rho>0$ sufficiently small such   
that $\overline{\psi}_\rho=\psi_\rho+\rho|x|^2$ satisfies   
\begin{equation}\nonumber   
\det\overline{\psi}_{ij}<\mu_0\quad\mbox{in~}\Omega,   
\end{equation}   
where we dropped the index $\rho$ as $\rho$ is fixed now.   
\end{remark}   
   
\begin{lemma}   
 For $\overline{\psi}$ as constructed above, $u\le\overline{\psi}$   
is valid during the evolution.   
\end{lemma}   
\begin{proof}   
The function $\overline{\psi}$ satisfies the elliptic differential   
inequality   
\begin{equation}\nonumber   
\left\{\begin{array}{rcll}   
\det\overline{\psi}_{ij}&<&\mu_0&\mbox{in~}\Omega,\\[0.2em]   
\overline{\psi}_\nu&=&   
\varphi\left(x,\overline{\psi}\right)&\mbox{on~}\partial\Omega  
\end{array}\right.   
\end{equation}   
and the parabolic differential inequality   
\begin{equation}\nonumber   
\left\{\begin{array}{rcll}   
\dot{\overline{\psi}}&>&\Phi\left(\log\det\overline{\psi}_{ij}  
-\log\mu_0\right)   
&\mbox{in~}\Omega\times[0,T),\\[0.2em]   
\overline{\psi}_\nu&=&\varphi\left(x,\overline{\psi}\right)&   
\mbox{on~}\partial\Omega\times[0,T)   
\end{array}\right.   
\end{equation}   
as $\overline{\psi}$ is independent of $t$,    
so $\dot{\overline{\psi}}=0$.   
 Furthermore we have the following elliptic differential inequality   
\begin{equation}\nonumber   
\left\{\begin{array}{rcll}   
\det(u_0)_{ij}&\ge&\mu_0   
&\mbox{in~}\Omega,\\[0.2em]   
(u_0)_\nu&=&\varphi(x,u_0)&\mbox{on~}\partial\Omega   
\end{array}\right.  
\end{equation}   
and the parabolic differential inequality   
\begin{equation}\nonumber   
\left\{\begin{array}{rcll}   
\dot u&=&\Phi\left(\log\det u_{ij}-\hat f\right)\\[0.2em]   
  &\le&\Phi\left(\log\det u_{ij}-\log\mu_0\right)  
&\mbox{in~}\Omega\times[0,T),\\[0.2em]   
u_\nu&=&\varphi(x,u)&\mbox{on~}\partial\Omega\times[0,T).   
\end{array}\right.   
\end{equation}   
We combine the elliptic differential   
 inequalities and obtain by the mean   
value theorem with a positive definite matrix $a^{ij}$   
and a positive function $C$  
\begin{equation}\nonumber   
\left\{\begin{array}{rcll}   
a^{ij}\left(u_0-\overline\psi\right)_{ij}  
&>&0&\mbox{in~}\Omega,\\[0.2em]   
\left(u_0-\overline\psi\right)_\nu  
&=&C\cdot\left(u_0-\overline\psi\right)&   
\mbox{on~}\partial\Omega,   
\end{array}\right.   
\end{equation}   
thus we obtain $u=u_0\le\overline\psi$ for $t=0$ in view of the  
elliptic maximum principle. From the parabolic differential   
inequalities we get   
\begin{equation}\nonumber   
\left\{\begin{array}{rcll}   
\dot u-\dot{\overline{\psi}}&<&  
\tilde a^{ij}\left(u-\overline\psi\right)_{ij}   
&\mbox{in~}\Omega\times[0,T),\\[0.2em]   
\left(u-\overline\psi\right)_\nu  
&=&\tilde C\cdot\left(u-\overline\psi\right)&   
\mbox{on~}\partial\Omega\times[0,T),   
\end{array}\right.   
\end{equation}   
so the parabolic maximum principle gives $u\le\overline\psi$   
for all $t\ge0$.   
\end{proof}   
   
\begin{corollary}   
\label{introduce delta}  
 For $\psi$ as constructed above there exists a positive   
constant $\delta_0$ such that   
\begin{equation}\nonumber   
(\psi-u)_\nu\ge\delta_0>0.   
\end{equation}  
\end{corollary}   
\begin{proof}   
As $u\le\overline{\psi}$ we deduce from the Neumann boundary   
condition   
\begin{equation}\nonumber   
\overline{\psi}_\nu-u_\nu=\varphi\left(x,\overline{\psi}\right)   
-\varphi(x,u)=\int\limits_0^1\varphi_z\left(x,\tau\overline\psi   
+(1-\tau)u\right)\,d\tau\cdot\left(\overline\psi-u\right),   
\end{equation}   
so $\left(\overline{\psi}-u\right)_\nu\ge0$, and furthermore   
\begin{equation}\nonumber   
(\psi-u)_\nu=\left(\overline\psi-\rho|x|^2-u\right)_\nu\ge -2\rho   
\langle x,\nu\rangle\ge\delta_0>0   
\end{equation}  
as $\Omega$ is strictly convex and $0\in\Omega$.  
\end{proof}  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
  
\subsection{Interior estimates}  
  
To establish a priori $C^2$-estimates everywhere, we proceed  
as in \cite{ltu}. For the reader's convenience, however, we  
repeat the argument given there modified for the parabolic  
case. We may take $T$ slightly smaller than the maximal  
time interval for which a solution exists.  
We define for $(x,\xi,t)\in\overline\Omega  
\times S^{n-1}\times[0,T]$  
\begin{equation}\nonumber  
W(x,\xi,t):=\log w+\beta\left(u^iu_i+M(\psi-u)\right)  
\end{equation}  
where  
\begin{eqnarray}\nonumber  
w(x,\xi,t)&=&u_{\xi\xi}-2\langle\xi,\nu\rangle\left(\xi^i  
-\langle \xi,\nu\rangle\nu^i\right)\cdot\left(\varphi_i+  
\varphi_zu_i-u_k\nu^k_i\right)\\  
&\equiv&u_{\xi\xi}+a^ku_k+b,\nonumber  
\end{eqnarray}  
and $\nu$ is a smooth extension of the inner unit normal to  
$\partial\Omega$ that vanishes outside a tubular neighborhood  
of $\partial\Omega$; $a^k$ and $b$ depend only on $x$ and $u$.  
  
\begin{lemma}[Interior $C^2$-estimates]  
\label{interior lemma}  
 For a solution of the flow equation (\ref{flow equation}),   
$W$ attains its maximum over  
$\overline\Omega\times S^{n-1}\times[0,T]$ at a boundary point,   
i.\ e.\ in $\partial\Omega\times S^{n-1}\times[0,T]$,  
provided $\beta\gg M\gg 1$ are  
chosen large enough or $\left|D^2u\right|$   
is a priori bounded by a constant  
determined by the $C^2$-norm of $u_0$ and known or estimated  
quantities.  
\end{lemma}  
\begin{remark}  
More precisely we assume for the maximum of $W$ $w\ge 1$,  
$c\le\Phi'\frac  
{\lambda}{2}\tru$, see (\ref{drei}), and furthermore  
(\ref{gross eins}), (\ref{gross zwei}), where $\varepsilon$  
is determined just above (\ref{diamond}) and $\beta$ is  
determined directly below (\ref{diamond}). This gives a  
possibility to calculate an upper bound of $\left|D^2u\right|$ in view  
of the above a priori estimates, if the maximum of $W$ is  
attained in $\Omega\times S^{n-1}\times(0,T]$.  
\end{remark}  
\begin{proof}[Proof of Lemma \ref{interior lemma}.]   
We assume that $W$ attains its maximum in the point   
$(x_0,\xi_0,t_0)\in  
\Omega\times S^{n-1}\times(0,T]$  (but later on we write   
again $\xi$ for  
simplicity) and $w$ is positive in a neighborhood  
of $x_0$, so we calculate there  
\begin{eqnarray}  
W_i&=&\frac{w_i}{w}+2\beta u^ku_{ki}+\beta M(\psi-u)_i,\nonumber\\  
W_{ij}&=&\frac{w_{ij}}{w}-\frac{w_iw_j}{w^2}   
+2\beta u^k_ju_{ki}+2\beta u^ku_{kij}+\beta M(\psi-u)_{ij},\nonumber\\   
\dot W&=&\frac{\dot w}{w}+2\beta u^k\dot u_k+\beta M   
(\dot\psi-\dot u).\nonumber  
\end{eqnarray}   
We differentiate the flow equation twice  
\begin{eqnarray}  
\label{flow diff one}  
\dot u_i&=&\Phi'u^{kl}u_{kli}-\Phi'D_i\hat f,\\  
\dot u_{\xi\xi}&=&\Phi'u^{ij}u_{ij\xi\xi}-\Phi'u^{ik}u^{jl}  
u_{ij\xi}u_{kl\xi}-\Phi'D_{\xi\xi}\hat f\nonumber\\  
&&{}+\Phi''\left(u^{ij}u_{ij\xi}\right)^2  
-2\Phi''u^{ij}u_{ij\xi}D_\xi\hat f+\Phi''\left(D_\xi  
\hat f\right)^2\nonumber\\  
\label{flow diff two}  
&\le&\Phi'u^{ij}u_{ij\xi\xi}-\Phi'u^{ik}u^{jl}  
u_{ij\xi}u_{kl\xi}-\Phi'D_{\xi\xi}\hat f,  
\end{eqnarray}  
where we have used the concavity of $\Phi$.  
$D_\cdot$ indicates that the  
chain rule has not yet been applied to the respective terms.  
\par  
As $|u^ku_{k\nu}|$ is bounded on $\partial\Omega$ we may fix  
$M$ such that  
\begin{equation}  
\label{choice M}  
M\delta_0\ge 2\left.|u^ku_{k\nu}|  
\right|_{\partial\Omega},  
\end{equation}  
where we use $\delta_0$ as introduced in Corollary  
\ref{introduce delta}.  
\par  
Now we restrict our attention to the point where the maximum is  
attained. We have there $W_i=0$, $W_{ij}\le0$, $\dot W\ge0$ and  
$\Phi'>0$, so we get  
\begin{eqnarray}  
0&\le&\dot W-\Phi'u^{ij}W_{ij}\nonumber\\  
&\le&\frac{\dot w}{w}-\Phi'\frac{1}{w}u^{ij}w_{ij}  
+\Phi'\frac{1}{w^2}u^{ij}w_iw_j\label{eins}\\  
&&{}+2\beta u^k\dot u_k-2\beta\Phi'\Delta u-2\beta\Phi'u^k  
u^{ij}u_{ijk}-\lambda\beta M\Phi'\tru+c\beta M\nonumber  
\end{eqnarray}  
with $\lambda>0$ as in Remark \ref{lambda remark}.  
We remark that $c$ also depends on $\Phi'$.  
 From (\ref{flow diff one}) and (\ref{flow diff two})  
we get  
\begin{eqnarray}  
\dot w-\Phi'u^{ij}w_{ij}&\le&\dot u_{\xi\xi}+a^k\dot u_k  
-\Phi'u^{ij}u_{\xi\xi ij}  
-\Phi'a^ku^{ij}u_{kij}+c\cdot\trup\nonumber\\  
&\le&-\Phi'u^{ik}u^{jl}u_{ij\xi}u_{kl\xi}  
-\Phi'D_{\xi\xi}\hat f\nonumber\\&&{}+c\cdot\trupD.\label{zwei}  
\end{eqnarray}  
We assume now that $W$ is large in its maximum,  
more precisely $c\le\Phi'\frac{\lambda}{2}\tru$, combine  
(\ref{eins}) and (\ref{zwei}) and take (\ref{flow diff one})  
into account  
\begin{eqnarray}\label{drei}  
0&\le&-\Phi'u^{ik}u^{jl}u_{ij\xi}u_{kl\xi}  
-\Phi'D_{\xi\xi}\hat f  
+\Phi'\frac{1}{w}u^{ij}w_iw_j-2\beta w\Phi'u^kD_k  
\hat f\nonumber\\  
&&{}-2\beta w\Phi'\Delta u  
-\frac{\lambda}{2}\beta Mw\Phi'\tru  
+c\cdot\left(1+\left|D^2u\right|+\tru\right).  
\end{eqnarray}  
We consider the quantity $u^{ij}w_iw_j$ separately and use  
Young's inequality for $0<\varepsilon<1$ to be fixed later  
\begin{eqnarray}\label{vier}  
u^{ij}w_iw_j&=&u^{ij}(u_{\xi\xi i}+D_ia^ku_k+D_ib+a^ku_{ki})  
\cdot\nonumber\\  
&&{}\cdot  
(u_{\xi\xi j}+D_ja^lu_l+D_jb+a^lu_{lj})\nonumber\\  
&\equiv&u^{ij}(u_{\xi\xi i}+B_i+a^ku_{ki})  
(u_{\xi\xi j}+B_j+a^lu_{lj})\nonumber\\  
&\le&(1+\varepsilon)u^{ij}u_{\xi\xi i}u_{\xi\xi j}  
+\frac{2}{\varepsilon}u^{ij}B_iB_j  
+\frac{2}{\varepsilon}u^{ij}u_{kj}a^ku_{li}a^l\nonumber\\  
&&{}+u^{ij}B_iB_j+2B_ia^i+a^ia^ju_{ij}\nonumber\\  
&\le&(1+\varepsilon)u^{ij}u_{\xi\xi i}u_{\xi\xi j}  
+\frac{c}{\varepsilon}\left(1+\left|D^2u\right|+\tru\right).  
\end{eqnarray}  
On the other hand we get in view of $W_i=0$  
\begin{equation}\label{fuenf}  
u^{ij}w_iw_j\le c\beta^2w^2\left(1+\left|D^2u\right|+\tru\right),  
\end{equation}  
where $c$ depends on the constant $M$ fixed above.  
\par  
We assume that $u_{\xi\xi}$ and the greatest eigenvalue of  
$u_{ij}$ at $x_0$, $u_{\eta\eta}$, are nearly as large as $w$,  
more precisely  
\begin{equation}\label{gross eins}  
0<\frac{1}{1+\varepsilon}\le\frac{u_{\eta\eta}}{w}\le 1+\varepsilon,  
\end{equation}  
and for later use  
\begin{equation}\label{gross zwei}  
1\le u_{\xi\xi},\quad 1\le\left|D^2u\right|,\quad   
\frac{1}{2}\left|D^2u\right|\le w\le 2\left|D^2u\right|,  
\end{equation}  
so we get for $0<\varepsilon\ll 1$ in view of  
(\ref{vier}) and (\ref{fuenf})  
\begin{eqnarray}\label{sechs}  
\frac{1}{w}u^{ij}w_iw_j&=&(1-3\varepsilon)\frac{1}{w}u^{ij}w_iw_j  
+3\varepsilon\frac{1}{w}u^{ij}w_iw_j\\  
&\le&u^{\eta\eta}u^{ij}u_{\xi\xi i}u_{\xi\xi j}\nonumber\\  
&&{}+\frac{c}{\varepsilon w}\left(1+\left|D^2u\right|+\tru\right)  
+c\varepsilon\beta^2w\left(1+\left|D^2u\right|+\tru\right).\nonumber  
\end{eqnarray}  
We calculate for $-D_{\xi\xi}\hat f-2\beta wu^kD_k\hat f$  
in view of $W_i=0$  
\begin{eqnarray}  
-\Phi'D_{\xi\xi}\hat f  
-2\beta\Phi'wu^kD_k\hat f&\le&-\Phi'\hat f_{p_k}u_{k\xi\xi}  
-2\beta\Phi'wu^k\hat f_{p_i}u_{ik}\nonumber\\  
&&{}+c\beta\left(1+\left|D^2u\right|\right)  
+c\left(1+|D^2u|^2\right)\nonumber\\  
&\le&c\beta\left(1+\left|D^2u\right|\right)  
+c\left(1+\left|D^2u\right|^2\right),\nonumber  
\end{eqnarray}  
where it is important to notice that the $2\beta w\hat f_{p_i}  
u^ku_{ki}$-terms cancel. We plug this estimate and (\ref{sechs})  
in (\ref{drei})  
\begin{eqnarray}\nonumber  
0&\le&-\Phi'u^{ik}u^{jl}u_{ij\xi}u_{kl\xi}  
+\Phi'u^{\eta\eta}u^{ij}  
u_{\xi\xi i}u_{\xi\xi j}-2\beta w\Phi'\Delta u  
-\frac{\lambda}{2}w\beta M\Phi'\tru\\  
&&{}+\frac{c}{\varepsilon w}\left(1+\left|D^2u\right|+\tru\right)  
+c\varepsilon\beta^2 w\left(1+\left|D^2u\right|+  
\tru\right)\nonumber\\  
&&{}+c\beta\left(1+\left|D^2u\right|\right)  
+c\left(1+\left|D^2u\right|^2+\tru\right).\nonumber  
\end{eqnarray}  
The sum of the first two terms is known to be nonpositive,  
see e.\ g.\ \cite{ltu}.  
We choose $\varepsilon=\frac{1}{\beta^2}$, so we obtain  
\begin{eqnarray}\label{diamond}  
0&\le&-2\Phi'\beta\left|D^2u\right|^2  
-\frac{\lambda}{2}\left|D^2u\right|\beta M\Phi'\tru\\  
&&{}+c\beta^2\left(\left|D^2u\right|+\tru\right)  
+c\,\left|D^2u\right|\cdot  
\left(\left|D^2u\right|+\tru\right).\nonumber  
\end{eqnarray}  
If we fix $\beta$ sufficiently large, it is easy to see that  
$\left|D^2u\right|(x_0,t_0)$ has to be   
a priori bounded by a constant.  
\end{proof}  
  
\subsection{Remaining boundary estimates}  
  
The proof of the tangential $C^2$-estimates at the boundary can   
be carried out as in \cite{ltu}. There, however, the authors only  
mention that this estimate can be obtained similar as at the beginning  
of Section 3 there. So we repeat the argument for readers not  
familiar with \cite{ltu}.   
  
Before stating the lemma we wish to point out that it is in  
general not true that $\xi_0$ is a direction tangential to   
$\partial\Omega$ when $W$ attains its maximum at $(x_0,\xi_0,t_0)  
\in\partial\Omega\times S^{n-1}\times(0,T]$.  
  
\begin{lemma}  
The second derivatives of a solution $u$ of our flow  
equation (\ref{flow equation}) are a priori bounded in  
$\overline{\Omega}\times[0,T]$.  
\end{lemma}  
\begin{proof}  
In view of Lemma \ref{Double normal lemma} and Lemma   
\ref{interior lemma} we  may assume without loss of generality  
that $W$ attains its maximum at a point $(x,\xi,t)\in\partial  
\Omega\times S^{n-1}\times(0,T)$   
with $\xi\not=\nu$ and distinguish two cases.   
\par  
{\it (i) tangential:\/}  
If $\xi$ is tangential to $\partial\Omega$, we differentiate   
\begin{equation}\nonumber  
\nu^iu_i=\varphi(x,u)  
\end{equation}  
with respect to tangential directions under the assumptions  
stated in the proof of Lemma \ref{mixed lemma} and get in view  
of $D\omega(\hat x_0)=0$  
\begin{eqnarray}\nonumber  
&&{}\nu^i_{\xi\xi}u_i+2\nu^i_\xi u_{i\xi}+\nu^iu_{i\xi\xi}+\nu^i  
u_{in}\omega_{\xi\xi}\\  
&=&\varphi_{\xi\xi}+\varphi_n\omega_{\xi\xi}  
+2\varphi_{z\xi}u_\xi+\varphi_{zz}u_\xi u_\xi  
+\varphi_zu_{\xi\xi}+\varphi_zu_n\omega_{\xi\xi},\nonumber  
\end{eqnarray}  
so we obtain  
\begin{eqnarray}\label{boundary twice}  
u_{\nu\xi\xi}&\ge&-2\nu^i_\xi u_{i\xi}+\varphi_zu_{\xi\xi}-c\\  
&\ge&\varphi_zu_{\xi\xi}-c\nonumber  
\end{eqnarray}  
as $\partial\Omega$ is strictly convex.  
On the other hand the maximality of $W$ at $x$ gives $0\ge W_\nu$,  
\begin{equation}\nonumber  
0\ge u_{\xi\xi\nu}-c+w\beta\left(2u^iu_{i\nu}+M(\psi-u)_\nu\right)  
\end{equation}  
and furthermore using (\ref{choice M})  
and Corollary \ref{introduce delta}  
\begin{equation}\nonumber  
0\ge u_{\xi\xi\nu}-c,  
\end{equation}  
so we obtain in view of (\ref{boundary twice}) and $\varphi_z\ge  
c_\varphi>0$ the desired estimate $u_{\xi\xi}\le c$.  
\par  
{\it (ii) non-tangential:\/}  
If $\xi$ is neither tangential nor normal we need the tricky  
choice of $w$ in \cite{ltu}. We find $0<\alpha<1$  
and a tangential direction $\tau$ such that  
\begin{equation}\nonumber  
\xi=\alpha\tau+\sqrt{1-\alpha^2}\nu.  
\end{equation}  
We rewrite $w$ as  
\begin{eqnarray}\nonumber  
w(x,\xi)&=&u_{\xi\xi}-2\alpha\sqrt{1-\alpha^2}  
\tau^i(\varphi_i+\varphi_zu_i  
-u_k\nu^k_i)\\  
&=&u_{\xi\xi}-2\alpha\sqrt{1-\alpha^2} u_{\tau\nu}\nonumber  
\end{eqnarray}  
in view of the differentiated Neumann boundary condition, so  
we see that  
\begin{eqnarray}\nonumber  
u_{\xi\xi}&=&\alpha^2u_{\tau\tau}+(1-\alpha^2)u_{\nu\nu}  
+2\alpha\sqrt{1-\alpha^2} u_{\tau\nu}\\  
&=&\alpha^2u_{\tau\tau}+(1-\alpha^2)u_{\nu\nu}-w(x,\xi)+u_{\xi\xi}  
\nonumber  
\end{eqnarray}  
and obtain in view of the maximality of $W$ and the fact that  
$W-\log w$ is independent of $\xi$ and   
$w(x,\tau)=u_{\tau\tau}$, $w(x,\nu)=u_{\nu\nu}$  
  
\begin{eqnarray}\nonumber  
w(x,\tau)&\le&w(x,\xi),\\  
w(x,\xi)&=&\alpha^2u_{\tau\tau}  
+(1-\alpha^2)u_{\nu\nu}\nonumber\\  
&=&\alpha^2w(x,\tau)+(1-\alpha^2)w(x,\nu)\nonumber\\  
&\le&\alpha^2w(x,\xi)+(1-\alpha^2)w(x,\nu).\nonumber  
\end{eqnarray}  
Therefore $w(x,\xi)\le w(x,\nu)$ gives the upper bound  
$u_{\xi\xi}\le c$ proving the statement.  
\end{proof}  
  
In the following sections we consider the second boundary  
value problem. In section \ref{Proof of the main theorem   
section} we will come back to   
Neumann boundary conditions. Sections \ref{Curvature   
functions and the Legendre transformation section} to  
\ref{C2-estimates section}  
will not be used for the proof of Theorem \ref{main 1}.  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\section{Legendre transformation and Hessian functions}  
\label{Curvature functions and the Legendre transformation  
section}  
  
 
  
We introduce some classes of Hessian functions similar  
to \cite{cgiumj2000, Minkowski space}. A slightly different  
class of Hessian functions is considered in \cite{Urbas   
Hessian}.  
  
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  
$$F_i=\fracp{F}{\lambda^i}>0;$$  
then, $F$ can also be viewed as a function defined on the space  
of symmetric, positive definite matrices $Sym^+(n)$, for, let  
$(u_{ij})\in Sym^+(n)$ with eigenvalues $\lambda_i$, $1\le i\le n$,  
then define $F$ on $Sym^+(n)$ by  
$$F(u_{ij})=F(\lambda_i).$$  
We have  
$F\in C^{\infty}(Sym^+)\cap C^0\left(\ol{Sym^+}\right)$.  
If we define  
$$F^{ij}=\fracp{F}{u_{ij}},$$  
then we get in an appropriate coordinate system  
$$F^{ij}\xi_i\xi_j=\fracp{F}{\lambda_i}\left|\xi^i\right|^2  
  \quad\forall\,\xi\in\R{}^n,$$  
and $F^{ij}$ is diagonal, if $u_{ij}$ is diagonal.  
We define furthermore  
$$F^{ij,kl}=\frac{\partial^2 F}{\partial u_{ij}\partial u_{kl}}.$$  
  
\begin{definition}\label{DefinitionClassK}  
A Hessian 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}{\lambda_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 u^{jl}\eta_{ij}  
  \eta_{kl}\quad\forall\,\eta\in Sym,  
\end{equation*}  
where   
$\left(\tilde u^{ij}\right)$ denotes the   
inverse of $(u_{ij})$, or, equivalently,  
if we set $\hat F=\log F$,  
$$\hat F^{ij,kl}\eta_{ij}\eta_{kl}\le  
  -\hat F^{ik}\tilde u^{jl}\eta_{ij}  
  \eta_{kl}\quad\forall\,\eta\in Sym,$$  
where $F$ is evaluated at $(u_{ij})$.  
  
If $F$ satisfies  
$${\exists\,\varepsilon_0>0:\quad  
  \varepsilon_0 FH\equiv\varepsilon_0 F\,\tr u_i^j\le  
  F^{ij}u_{ik}u^k_j\label{EpsKStar}}$$  
for any $(u_{ij})\in Sym^+$, 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 Hessian functions $F$ which fulfill, instead of the  
homogeneity condition, the following weaker assumption  
$${\label{WeakenedHomogeneityCondition}  
  \exists\,\delta_0>0:\quad 0<\frac{1}{\delta_0}F\le  
  \sum\limits_iF_i\lambda_i\le\delta_0F}$$  
is denoted by an additional tilde, $F\in\left(\tilde K\right)$  
or $F\in\left(\tilde K^\star\right)$.  
  
A Hessian 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 Hessian functions of the class   
$\left(\tilde K^\star\right)$ as given in \cite{cgiumj2000,  
Minkowski space}.  
  
Let $H_k$ be the $k$-th elementary symmetric polynomials,  
\begin{eqnarray}  
H_k(\lambda_i)&:=&\sum\limits_{1\le i_1<\ldots<i_k\le n}  
  \lambda_{i_1}\cdot\ldots\cdot\lambda_{i_k},\quad 1\le k\le n,\\  
  \sigma_k&:=&\left(H_k\right)^\frac{1}{k}\nonumber  
\end{eqnarray}  
the respective Hessian functions homogeneous of degree $1$  
and define furthermore  
\begin{equation}\nonumber  
\tilde\sigma_k(\lambda_i):=\frac{1}{\sigma_k  
  \left(\lambda_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{cgiumj2000},   
\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 $F_{(i)}\in(\tilde K)$, and we may even allow  
$F_{(i)}\not=0$ on $\partial\Gamma_+$.   
  
An additional construction gives inhomogeneous   
examples \cite{Minkowski space}.  
Let $F$ be as in (\ref{K times example}),  
$\eta\in C^{\infty}(\R{}_{\ge0})$  
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(\lambda_i):=F\left(\exp\left(\int\limits_1^{\lambda_i}  
\frac{\eta(\tau)}{\tau}\,d\tau\right)\right)  
\end{equation}  
belongs to the class $\left(\tilde K^\star\right)$.  
\end{example}  
  
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. $\left(\tilde K^\star\right)\subset(\tilde K)\cap  
(CNS)$, moreover, when $F\in\left(\tilde K\right)\cap(CNS)$,  
$0<\frac{1}{c}\le F\le c$, and   
\begin{equation}\nonumber  
0<\lambda_1\le\ldots\le\lambda_n,  
\end{equation}  
then the following three conditions are equivalent  
\begin{equation*}  
\lambda_1\to0,\quad  
\lambda_n\to\infty,\quad  
{\rm tr\,}F^{ij}\to\infty.  
\end{equation*}  
\end{lemma}  
\begin{proof}  
We refer to \cite{Minkowski space}.  
\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<\lambda_1\le\ldots\le\lambda_n,  
\end{equation}  
and $0<\frac{1}{c}\le F\le c$. Then the following  
three conditions are equivalent  
$$\lambda_1\to0,\quad  
\lambda_n\to\infty,\quad  
{\rm tr\,}F^{*ij}\to\infty,$$
where $F^*$ is defined by
$$F^*(\lambda_i)=\frac{1}{F\left(\frac{1}{\lambda_i}\right)}.$$  
\end{lemma}  
\begin{proof}  
We have $F_1\ge\ldots\ge F_n>0$, see \cite{cgmz1997,   
Urbas Hessian}, so we get in view of the definition  
of $F^*$  
$$F^*_i(\lambda_1,\ldots,\lambda_n)=  
\frac{F_i\left(\frac{1}{\lambda_i}\right)}{F^2}  
\cdot\frac{1}{\lambda_i^2}.$$  
Thus $F^*_1\to\infty$ as $\lambda_1\to0$ gives the result  
and $\lambda_n\to\infty$ forces $\lambda_1\to0$   
in view of Lemma \ref{three equivalences}.  
To get $\tr F^{*ij}\to\infty$, $\lambda$ 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\lambda_i^2\le(c(\varepsilon)+\varepsilon\cdot  
|\lambda|)\cdot\sum\limits_iF_i,  
\end{equation}  
provided $0<\frac 1c\le F\le c$.
\end{lemma}  
\begin{proof}  
See \cite{Urbas Hessian}.  
\end{proof}  
  
Instead of Lemma \ref{three equivalences for F*}   
we get the following  
weaker result for $S_{n,n-k}^*$.  
\begin{lemma}\label{relaxed three equivalences}  
Let $F=(S_{n,n-k})^*=H_{k}$, $1\le k\le n-1$, and assume  
$0<\frac{1}{c}\le F\le c$ and   
\begin{equation}\nonumber  
0<\lambda_1\le\ldots\le\lambda_n,  
\end{equation}  
then   
$$F_1\ge\ldots\ge F_n>0.$$  
Moreover, at least one of the following conditions is fulfilled  
\begin{equation}\label{easy compact case}  
 F_n\ge\frac{1}{c}>0  
\end{equation}  
or  
$$\tr F^{ij}\to\infty.$$   
\end{lemma}  
\begin{proof}  
The first inequality and the case $k=1$ are obviously true.   
If $0<\frac{1}{c}\le\lambda_{n-k+1}\le\ldots\le\lambda_{n}$,   
then $F_n\ge\frac{1}{c}>0$. If $\lambda_{n-k+1}\to0$, then  
$\lambda_n\to\infty$ as $\lambda_{n-k+1}\cdot\ldots\cdot  
\lambda_n\ge\frac{1}{c}\cdot H_k\ge\frac{1}{c}>0$, so  
\begin{eqnarray*}  
(H_k)_1&\ge&\ldots\ge(H_k)_{n-k+1}\ge  
\lambda_{n-k+2}\cdot\ldots\cdot\lambda_n  
=\frac{1}{\lambda_{n-k+1}}\cdot  
\lambda_{n-k+1}\cdot\ldots\cdot\lambda_n\\  
&\ge&\frac{1}{\lambda_{n-k+1}}\cdot\frac{1}{c}\cdot H_k  
\ge\frac{1}{\lambda_{n-k+1}}\cdot\frac{1}{c}\to\infty.  
\end{eqnarray*}  
\end{proof}  
By direct calculations we obtain the following  
\begin{lemma}  
If $u$ is a strictly convex $C^2$-solution of   
(\ref{flow equation second}),  
then the Legendre transform of  
$u$, $u^*:\Omega^*\times[0,T)\to\R{}$, defined by  
$$u^*(y,t):=x^iu_i(x,t)-u(x,t)  
\equiv x^iy_i-u,\quad y^i=u^i(x,t)$$  
satisfies the evolution equation  
$$\left\{\begin{array}{rcll}  
\dot u^*&=&\hat F^*(D^2u^*)-\log g^*(y,u^*,Du^*)&\mbox{in~}  
\Omega^*\times[0,T),\\  
Du^*(\Omega^*)&=&\Omega,&\\  
\left.u^*\right|_{t_0}&=&u_0^*&\mbox{in~}\Omega^*,  
\end{array}\right.$$  
where $u^*_0$ is defined similarly as $u^*$,  
$$F^*(\lambda_i)=\frac{1}{F\left(\frac{1}{\lambda_i}\right)},$$  
and  
$$g^*(y,z^*,q^*):=\frac{1}{g(q^*,y^iq_i^*-z^*,y)}$$  
and the time derivative of $u^*$ is taken with $y$ fixed.  
\end{lemma}  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\section{Strict obliqueness}\label{Strict obliqueness section}  
  
\begin{lemma}\label{Strict obliqueness lemma}  
As long as a solution as in Theorem \ref{main theorem} 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}  
\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}  
  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\section{$\dot u$- and $C^0$-estimates}  
\label{C0-estimates section}  
  
\begin{remark}[$\dot u$-estimates]  
The results of section \ref{section 2} hold also for  
the flow (\ref{flow equation second}), as in both cases,  
the flow equation is parabolic and the boundary condition  
is strictly oblique.   
\end{remark}  
  
If condition (\ref{g condition}) is fulfilled, uniform  
$C^0$-a priori estimates follow immediately by integrating  
the estimate in Lemma \ref{dot u bounded}, see also  
Lemma \ref{lower estimates}. In the case of  
condition (\ref{dot u positive}), the positivity of $\dot u$,  
Lemma \ref{u-dot remains positive}, gives a lower bound for $u$.  
So it remains to establish an upper bound for $u$ in the  
case $\dot u\ge0$.  
  
\begin{lemma}  
A solution $u$ of our flow equation   
(\ref{flow equation second})  
is uniformly bounded.  
\end{lemma}  
\begin{proof}  
The concavity of $\hat F(\cdot)$ gives the estimate  
$$\hat F(D^2u)\le \hat F^{ij}(1,\,\ldots,\,1)(u_{ij}-\delta_{ij})  
+\hat F(1,\,\ldots,\,1)\le c\cdot\Delta u+c.$$  
For $0<t_1<t_2$ we integrate the flow equation and estimate  
in view of the inequality above, the divergence theorem and  
$|Du|\le c$ ($Du(\Omega)=\Omega^*$)  
\begin{eqnarray}  
\int\limits_{t_1}^{t_2}\int\limits_\Omega\log g(x,u,Du)  
&\le&c\cdot\int\limits_{t_1}^{t_2}\int\limits_\Omega\Delta u  
+c\cdot(t_2-t_1)-\int\limits_\Omega  
\left(\left.u\right|_{t_2}-\left.u\right|_{t_1}\right)  
\nonumber\\  
&\le&c\cdot\int\limits_{t_1}^{t_2}\int\limits_{\partial\Omega}|Du|  
+c\cdot(t_2-t_1)-\int\limits_\Omega  
\left(\left.u\right|_{t_2}-\left.u\right|_{t_1}\right)  
\nonumber\\  
&\le&c\cdot(t_2-t_1)  
-\int\limits_\Omega  
\left(\left.u\right|_{t_2}-\left.u\right|_{t_1}\right).  
\label{integrated flow equation}  
\end{eqnarray}  
The boundedness of $Du$ and the convexity of $\Omega$ yield  
the estimate  
\begin{equation}\label{difference bounded}  
|u(x_1,t)-u(x_2,t)|\le c_{\Omega,\Omega^*}  
\quad\forall\, x_1,\,x_2\in\Omega\quad\forall\, t>0.  
\end{equation}  
So we obtain from (\ref{integrated flow equation}) for any  
$x\in\Omega$  
\begin{equation}\label{int+est flow equation}  
\frac{1}{|\Omega|}  
\int\limits_{t_1}^{t_2}\int\limits_\Omega\log g(x,u,Du)  
\le c+c\cdot(t_2-t_1)-u(x,t_2)+u(x,t_1).  
\end{equation}  
Now we fix $T>0$ and assume that  
$$u(x_0,T)=\max\limits_{\overline{\Omega}\times[0,T]}u  
=:M>\max\left\{2\max\limits_{\overline{\Omega}}u_0,0\right\}.$$  
We choose $t\in(0,T)$ maximal such that $u(x_0,t)=  
\frac{M}{2}$. From the mo\-no\-to\-ni\-ci\-ty of $g$ and  
(\ref{int+est flow equation}) we get the estimate  
\begin{eqnarray*}  
\frac{M}{2}&=&u(x_0,T)-u(x_0,t)\\&\le& c+c\cdot(T-t)  
-(T-t)\cdot\inf\limits_{x\in\Omega}  
\inf\limits_{p\in\Omega^*}\log  
g\left(x,\frac{M}{2}-c_{\Omega,\Omega^*},p\right)  
\end{eqnarray*}  
and after rearranging  
$$\frac{\frac{M}{2}-c}{T-t}\le c  
-\inf\limits_{x\in\Omega}  
\inf\limits_{p\in\Omega^*}\log  
g\left(x,\frac{M}{2}-c_{\Omega,\Omega^*},p\right).$$  
For $M\to\infty$ the left-hand side of this inequality  
remains positive, whereas the right-hand side tends to  
$-\infty$ in view of (\ref{g asymptotics}), so $M$ is  
a priori bounded proving the lemma.  
\end{proof}  
  
\begin{corollary}  
During the evolution, $\hat F(D^2u)$ is a priori bounded  
from above and from below.  
\end{corollary}  
\begin{proof}  
This follows from $|Du|\le c$ and from the flow equation.  
\end{proof}  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\section{Strict obliqueness estimates}  
\label{Strict obliqueness estimates section}  
  
The following lemma establishes a uniform lower bound for  
the quantity whose positivity we proved in Lemma  
\ref{Strict obliqueness lemma}.  
\begin{lemma}\label{obliqueness estimates lemma}  
During the evolution (\ref{flow equation second}),  
we have the strict obliqueness estimate  
\begin{equation}\label{strict obliqueness estimate}  
\langle\nu(x),\nu^*(Du(x,t))\rangle\ge\frac{1}{c}>0,  
\quad x\in\partial\Omega,  
\end{equation}  
where $\nu$ and $\nu^*$ denote the inner unit normals of  
$\Omega$ and $\Omega^*$, respectively. The positive lower  
bound is independent of $t$.  
\end{lemma}  
\begin{proof}  
We assume that a solution of our flow equation exists for a  
time interval $(0,T]$ and prove an estimate for  
$h_{p_k}\nu^k$ during this time interval which is  
independent of $T$. To establish a 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 we define  
$$v=h_{p_k}\nu^k+Ah(Du).$$  
The function $\left.v\right|_{\partial\Omega\times(0,T]}$  
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&=&v_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 v.  
\end{eqnarray}  
We assume for a moment that there holds  
\begin{equation}\label{bound for v_n}  
v_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{Strict obliqueness lemma} 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 obtain at $x_0$  
$$Au_{kl}h_{p_k}h_{p_l}\ge-c(A)h_{p_n}+\frac{1}{c_1}$$  
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}).  
There is another slightly shorter proof of this inequality  
obtained by constructing a barrier in a tubular  
neighborhood of $\partial\Omega$ avoiding the term  
$|x-x_0|^2$ below, but we prefer the following proof as it  
uses only local properties of the involved quantities. As  
for a similar proof with more details we refer to  
Lemma \ref{Double normal lemma}. Direct calculations   
using (\ref{flow equation second}) give  
\begin{eqnarray*}  
Lv&\le&\hat F^{ij}u_{li}u_{jm}\nu^kh_{p_kp_lp_m}  
+A\cdot\hat F^{ij}u_{ki}u_{lj}h_{p_kp_l}+c(A)\cdot\trF\\  
&\le&c(A)\cdot\trF  
\end{eqnarray*}  
for $A$ sufficiently large and  
$$Lw:=-\dot w+\hat F^{ij}w_{ij}-\hat g_{p_i}w_i.$$  
We wish to mention that this definition differs from the 
definition of L in Lemma \ref{Double normal lemma} by a sign.
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\trF.  
\end{equation}  
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.  
As $v$ is bounded and attains its minimum over  
$\partial\Omega\times[0,T]$ in $(x_0,t_0)$ we find  
$C\gg B\gg1$ and an affine linear function $l$ with $l(x_0)=0$
such that the function  
$$\Theta:=C\cdot\vartheta+B\cdot|x-x_0|^2+v-v(x_0,t_0)+l$$  
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  
$$(C\cdot\vartheta+v+l)_n(x_0,t_0)\ge0$$  
as the function $C\cdot\vartheta+B\cdot|x-x_0|^2+v-v(x_0,t_0)+l$  
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 $|Dh^*|=1$ on  
$\partial\Omega$. We define  
$$v^*=h^*_{q_k}(Du^*)\nu^{*k}+Ah^*(Du^*)$$  
and a linear operator by  
$$L^*w:=-\dot w+\hat  F^{*ij}w_{ij}-\hat g^*_{q_i}w_i.$$  
As before we obtain that $\left.v^*\right|_{\partial\Omega  
\times[0,T]}$ is  
positive. We fix $T>0$ and assume that $\left.v^*\right|_%  
{\partial\Omega\times[0,T]}$ attains its minimum in $(y_0,  
t_0)$. As we wish to establish a positive lower bound for  
$v^*$ we may assume that $t_0>0$. By calculations as above  
-- using Lemma \ref{three equivalences for F*} --  
we obtain in $(y_0,t_0)$ 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  
$|Dh^*|=1$ on $\partial\Omega^*$, so we deduce  
$u^*_{kl}h^*_{q_k}h^*_{q_l}\ge\frac{1}{c}>0$. 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{$C^2$-estimates}\label{C2-estimates section}  
  
 For convenience we use the notation $h_{p_k}(Du)=\beta^k$.  
We state the following estimates on $\partial\Omega$ obtained  
by differentiating the boundary condition  
\begin{equation}\label{mixed derivatives vanish}  
u_{\tau\beta}=0,\quad u_{\nu\beta}\ge0  
\end{equation}  
where $\tau$ and $\nu$ denote a tangential vector and the  
inner unit normal, respectively, see also (\ref{h(Du)=0  
differentiated}). The estimates in this section are valid  
for any $\varepsilon>0$ if $\varepsilon$ is not fixed  
explicitly. Thus multiplying a term of the form  
$c(\varepsilon)+\varepsilon\cdot M$ with a constant yields  
again a term of the form $c(\varepsilon)+\varepsilon\cdot M$.  
We obtain the following  
  
\begin{lemma}\label{u beta beta lemma}  
A solution of our flow equation (\ref{flow equation second})   
in a time interval $[0,T]$  
satisfies for all $\varepsilon>0$  
\begin{equation}\label{u beta beta estimate}  
u_{\beta\beta}\le c(\varepsilon)+\varepsilon\cdot M  
\quad\mbox{in~}\Omega,  
\end{equation}  
where $M:=\sup\limits_{\overline{\Omega}\times[0,T]}  
|D^2u|$.  
\end{lemma}  
\begin{proof}  
We set $H=h(Du)$,  
$$Lw:=-\dot w+\hat F^{ij}w_{ij}-\hat g_{p_i}w_i$$  
and compute the differential inequality  
$$LH\ge-(c(\varepsilon)+\varepsilon M)  
\cdot\trF$$  
where we have used Lemma \ref{properties of F*...}  
and the boundedness of $\hat F(D^2u)$.  
Applying the maximum principle as in Lemma \ref{obliqueness  
estimates lemma} to the function  
$-A\cdot(c(\varepsilon)+\varepsilon\cdot M)\cdot\vartheta  
-B\cdot|x-x_0|^2+H+l$ with  
$\vartheta$, $l$ as in Lemma \ref{obliqueness estimates lemma},  
$A\gg B\gg 1$ sufficiently large positive constants  
and $x_0\in\partial\Omega$, we obtain  
\begin{equation}\label{u beta nu estimate}  
u_{\beta\nu}\le c(\varepsilon)+\varepsilon\cdot M\quad  
\mbox{on~}\partial\Omega.  
\end{equation}  
We remark that $\varepsilon$ in (\ref{estimate for theta}) 
is fixed and is not related to $\varepsilon$ used here.
In view of $u_{\beta\tau}=0$ on $\partial\Omega$  
and the strict obliqueness estimate or by using the  
maximum principle as above for $\beta$ instead  
of $\nu$, the claimed inequality follows.  
\end{proof}  
  
As to the interior second derivative estimates we recall  
from \cite{gt}  
\begin{lemma}\label{interior estimates lemma}  
For a solution of our flow equation in a time interval  
$[0,T]$ we have the estimate  
$$\sup\limits_{\Omega\times[0,T]}|D^2u|\le c  
+\sup\limits_{\partial\Omega\times[0,T]}|D^2u|  
+\sup\limits_{\Omega\times\{0\}}|D^2u|.$$  
\end{lemma}  
\begin{proof}  
Similar computations as in \cite{gt} in the elliptic  
case -- under the assumption that  
$$\overline{\Omega}\times[0,T]\times S^{n-1}\ni  
(x,t,\xi)\mapsto\gamma\cdot|Du(x,t)|^2  
+\log u_{\xi\xi}(x,t)$$  
attains its maximum in $\Omega\times(0,T]\times  
S^{n-1}$ for $\gamma$ sufficiently large -- give the above  
bound. We remark, that we used $F\in\left(\tilde F^\star\right)$  
and not only $F\in\left(\tilde K\right)\cap(CNS)$.  
\end{proof}  
  
Up to now we control $u_{\beta\tau}(=0)$, $u_{\beta\beta}$ and  
we have an interior estimate for the second derivatives  
of $u$. In the following lemma we bound double tangential  
derivatives at the boundary. This completes the $C^2$-a  
priori estimates.  
\begin{lemma}  
For a solution of our flow equation in a time interval  
$[0,T]$ the second tangential derivatives at the boundary  
are a priori uniformly bounded.  
\end{lemma}  
\begin{proof}  
We may assume  
\begin{equation}\label{choose u11 maximal}  
\sup\limits_{\partial\Omega\times[0,T]}\sup\limits_%  
{|\tau|=1,\,\langle\tau,\nu\rangle=0}u_{\tau\tau}  
=u_{11}(x_0,t_0)  
\end{equation}  
where $x_0\in\partial\Omega$, $t_0\in(0,T]$ and furthermore  
that $\nu=e_n$ is the inner unit normal at $x_0\in\partial  
\Omega$. At a boundary point we decompose any direction $\xi$,  
i.\ e.\ a vector $\xi\in\R{}^n$ such that $|\xi|=1$,  
$$\xi=\tau(\xi)+\frac{\langle\nu,\xi\rangle}{\langle\beta,  
\nu\rangle}\beta,$$  
where  
$$\tau(\xi)=\xi-\langle\nu,\xi\rangle\nu-\frac{\langle\nu,  
\xi\rangle}{\langle\beta,\nu\rangle}\beta^T,\quad  
\beta^T=\beta-\langle\beta,\nu\rangle\nu,$$  
and obtain the estimate  
\begin{equation}\label{tau norm estimate}  
|\tau(\xi)|^2\le1+c\cdot\langle\nu,\xi\rangle^2-2\langle\nu,  
\xi\rangle\frac{\langle\beta^T,\xi\rangle}{\langle\beta,  
\nu\rangle}.  
\end{equation}  
We set $\tau:=\tau(e_1)$ and obtain on $\partial\Omega$  
in view of the estimates (\ref{mixed derivatives vanish}),  
(\ref{u beta beta estimate}), (\ref{choose u11 maximal})  
and (\ref{tau norm estimate}) above  
$$u_{11}\le\left(1+c\cdot\langle\nu,e_1\rangle^2-2  
\frac{\langle\nu,e_1\rangle\cdot\langle\beta^T,e_1\rangle}%  
{\langle\beta,\nu\rangle}\right)\cdot u_{11}(x_0,t_0)  
+(c(\varepsilon)+\varepsilon\cdot M)\langle\nu,e_1\rangle^2.$$  
\par  
Before we proceed, we establish an estimate for the quantity  
$M$ introduced in Lemma \ref{u beta beta lemma}.  
Lemma \ref{interior estimates lemma} gives  
\begin{equation}\label{preliminary M estimate}  
M\le c+\sup\limits_{\partial\Omega\times[0,T]}  
|D^2u|,  
\end{equation}  
where the supremum also includes non-tangential directions.  
For a direction $\xi$ we obtain in view of $u_{\beta\tau}=0$  
on $\partial\Omega$, (\ref{choose u11 maximal}) and  
(\ref{u beta beta estimate})  
\begin{eqnarray*}  
u_{\xi\xi}&\le&u_{\tau(\xi)\tau(\xi)}  
+c\cdot u_{\beta\beta}\\  
&\le&c\cdot u_{11}(x_0,t_0)+c(\varepsilon)+\varepsilon\cdot M.  
\end{eqnarray*}  
Combining this inequality for $\varepsilon>0$ small enough  
with (\ref{preliminary M estimate}) we get  
\begin{equation}\label{estimate for M}  
M\le c\cdot(1+u_{11}(x_0,t_0)).  
\end{equation}  
We dropped $\varepsilon$ as it was fixed  
sufficiently small to get  
this inequality and will be fixed differently later-on.  
\par  
We may assume in view of (\ref{estimate for M})  
for the rest of the proof  
that $u_{11}(x_0,t_0)\ge 1$ and for  
$$w:=\frac{u_{11}}{u_{11}(x_0,t_0)}+2\frac{\langle\nu,  
e_1\rangle\cdot\langle\beta^T,e_1\rangle}{\langle\beta,  
\nu\rangle}$$  
we obtain --  
by using (\ref{estimate for M}) -- the estimate  
$$w\le 1+c(\varepsilon)|x'|^2\quad\mbox{on~}\partial\Omega  
\mbox{~near~}x_0,$$  
where $x'\equiv(x^1,\ldots,x^{n-1})$, and we get furthermore  
$w\le c(\varepsilon)$ everywhere on $\partial\Omega$.  
We consider $2\frac{\langle\nu,e_1\rangle\cdot\langle\beta^T,  
e_1\rangle}{\langle\beta,\nu\rangle}$ as a known function  
depending on $(x,Du)$, use the flow equation,  
and obtain in $\Omega$ by direct calculation  
$$-\dot w+\hat F^{ij}w_{ij}-\hat g_{p_i}w_i\ge-c\cdot(c(\varepsilon)+  
\varepsilon\cdot M)\,\trF.$$  
Thus the maximum principle gives with a barrier function as  
constructed above  
\begin{equation}\label{u 1 beta beta estimate}  
u_{11\beta}(x_0,t_0)\le(c(\varepsilon)+\varepsilon\cdot M)\,  
u_{11}(x_0,t_0).  
\end{equation}  
Differentiating the boundary condition twice in the direction  
$e_1$ we obtain at $(x_0,t_0)$  
$$h_{p_kp_l}u_{k1}u_{l1}+u_{\beta 11}  
+u_{\beta n}\omega_{11}=0,$$  
where $\omega$ is a function such that locally  
$\partial\Omega$ is represented as $\graph\omega$ over its  
tangent plane at $x_0$.  
Combining this equality with (\ref{u 1 beta beta estimate})  
and (\ref{u beta nu estimate}), we obtain in the non-trivial  
case $u_{11}(x_0,t_0)\ge c(\partial\Omega)$ which we will  
assume in the following  
\begin{equation}\label{first ingredient}  
(c(\varepsilon)+\varepsilon\cdot M)\cdot u_{11}(x_0,t_0)  
+h_{p_kp_l}u_{k1}u_{l1}\ge 0.  
\end{equation}  
Inequality (\ref{estimate for M}) and the uniform concavity  
of $h$ yield  
$$(c(\varepsilon)+\varepsilon\cdot u_{11}(x_0,t_0))\cdot  
u_{11}(x_0,t_0)\ge\frac{1}{c}(u_{11}(x_0,t_0))^2.$$  
We now fix $\varepsilon>0$ sufficiently small and get  
a bound for $u_{11}(x_0,t_0)$.  
\end{proof}  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\section{Proof of the main theorems}\label{section 4}  
\label{Proof of the main theorem section}  
We return to the case of a Neumann boundary value problem.  
  
 From the uniform $C^2$-estimates for $u$ and the uniform $C^0$-estimates for
$\dot u=\Phi$ we obtain that $u$ remains uniformly convex
and we 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} imply uniform bounds in  
$C^{k}$, for all $k\ge 0$. It follows that a smooth solution of  
(\ref{flow  
equation}) exists for all $t\ge 0$.  
We then need the following Lemma.  
\begin{lemma}\label{longtime->convergence}  
If a solution of the flow equation   
(\ref{flow equation}) exists for all $t\ge0$ and  
either (\ref{structure 2}) or (\ref{positive velocity})  
are satisfied, then  
the flow converges to a solution of the Neumann problem  
i.\ e.  
\begin{equation}\nonumber  
\lim\limits_{t\to\infty}u(x,t)=:u^\infty(x)  
\end{equation}  
exists and  
\begin{equation}\nonumber  
\left\{\begin{array}{rcll}  
u^\infty_\nu&=&\varphi(x,u^\infty)&\mbox{on~}\partial\Omega,\\  
\det u_{ij}^\infty&=&f(x,u^\infty,Du^\infty)&\mbox{in~}\Omega.  
\end{array}\right.  
\end{equation}  
Moreover, $u(t,\cdot)\to u^\infty$ smoothly. If   
(\ref{structure 2}) holds, then the convergence is exponentially  
fast in any $C^k$-norm, $k\ge0$.   
\end{lemma}  
\begin{proof}  
First, we assume that (\ref{positive velocity}) is fulfilled.  
We may assume $\dot u(0,\cdot)\not\equiv 0$ and proceed as in  
\cite{cgiumj2000}. Integrating the flow equation gives  
\begin{equation}\nonumber  
u(t,x)-u(0,x)=\int\limits_0^t\Phi.  
\end{equation}  
The left-hand side is uniformly bounded in view of the  
$C^0$-estimates. As $\log\det u_{ij}-\hat f$ is non-negative  
we find $t_k=t_k(x)\to\infty$ such that  
\begin{equation}\label{tk converge}  
\left.\left(\log\det(D^2u)-\hat f(x,u,Du)\right)\right|_{t=t_k}\to0.  
\end{equation}  
On the other hand, $u(x,\cdot)$ is monotone, so  
$\lim\limits_{t\to\infty}u(x,t)=:u^\infty(x)$ exists and  
is smooth in view of our a priori estimates. Dini's theorem  
and interpolation inequalities of the form  
\begin{equation}\nonumber  
||D\tilde u||\le c||\tilde u||\cdot(||D^2\tilde u||+||D\tilde u||),  
\end{equation}  
for $\tilde u=u-u^\infty$,  
where $||\cdot||$ denotes the sup-norm, yield smooth convergence  
$u\to u^\infty$. Thus we conclude in view of (\ref{tk converge})  
that $u^\infty$ is a smooth  
solution of the stationary problem (\ref{stationary problem}).  
  
In case (\ref{structure 2}) we use the a priori bounds for  
all $C^k$-norms and $\dot u\to0$, see Lemma~%  
\ref{lower bound for u-dot}, to get smooth convergence  
to $u^\infty(x)$.  
Again by Lemma~\ref{lower bound for u-dot} we conclude  
$$||u-u^\infty||<c_0 e^{-\lambda_0t}$$  
for constants $\lambda_0>0$, $c_0>0$.  
Then we apply interpolation inequalities as above to   
$\tilde u=u-u^\infty$ and derive   
$$||u-u^\infty||_k<c_k e^{-\lambda_kt}$$  
for any $k\ge 0$ and positive constants $\lambda_k$,   
$c_k$. Clearly, $u^\infty$ is a smooth  
solution of the stationary problem (\ref{stationary problem}).  
\end{proof}  
  
\begin{remark}  
By an iteration method applied to the interpolation  
inequality one can even show (Lemma \ref{Iteration}) that   
$\lambda_k$ can be chosen independent of $k$.  
\end{remark}  
  
\begin{proof}[Proof of Theorem \ref{main 1}   
and Theorem \ref{main theorem}]  
The a priori estimates obtained so far guarantee longtime  
existence for solutions of our flow equations, so   
the statement of Theorem \ref{main 1} follows from Lemma   
\ref{longtime->convergence} and the claim of Theorem  
\ref{main theorem} follows from a similar lemma.  
\end{proof}  
  
%%%%%%%%%%%%%%%%  
\begin{appendix}  
  
\section{Oblique boundary value problems}  
  
\subsection{Flows solving the oblique boundary value problem for  
  Hessian equations}  
  
We get the following theorem for Hessian flow equations.  
  
\begin{theorem}\label{Appendix Theorem}  
Let $\Omega\subset\R{n}$ be a smooth uniformly strictly convex  
domain, $f$ a positive smooth function defined on   
$\overline{\Omega}\times\R{}\times\R{n}$ with $f_z\ge 0$,  
let $\varphi$ be a smooth function defined on $\overline{\Omega}  
\times\R{}$ with $\varphi_z>0$   
in $\partial\Omega\times\R{}$ and   
\begin{equation}  
\varphi(x,z)\to\sigma\infty,\quad z\to\sigma\infty,\quad  
\sigma\in\{-1,1\},\nonumber  
\end{equation}  
 uniformly in $x$. Let $F\in\left(\tilde K^\star\right)$  
and $\beta$ a smooth vectorfield on $\partial\Omega$ that  
is $C^1$-close to the inner unit normal $\nu$ as  
described in \cite{Urbas Calc 1998}.   
Then the initial value problem for the parabolic boundary  
value problem  
\begin{equation}\nonumber  
\left\{\begin{array}{rcll}  
\dot u&=&\log F\left(D^2u\right)  
-\log f(x,u,Du)&\mbox{in~}\Omega,\\  
u_\beta&=&\varphi(x,u)&\mbox{on~}\partial\Omega  
\end{array}\right.  
\end{equation}  
has a unique smooth strictly convex solution $u$ and $u$ converges  
smoothly to a solution of the Neumann boundary value problem  
\begin{equation}\nonumber  
\left\{\begin{array}{rcll}  
F\left(D^2u\right)&=&f(x,u,Du)&\mbox{in~}\Omega,\\  
u_\beta&=&\varphi(x,u)&\mbox{on~}\partial\Omega  
\end{array}\right.  
\end{equation}  
if we start with a smooth strictly convex   
subsolution $u_0=\left.u\right|_{t=0}$, i.\ e.  
\begin{equation}\nonumber  
\left\{\begin{array}{rcll}  
0&\le&F\left(D^2u_0\right)-f(x,u_0,Du_0)&\mbox{in~}\Omega,\\  
(u_0)_\beta&=&\varphi(x,u_0)&\mbox{on~}\partial\Omega  
\end{array}\right.  
\end{equation}  
and assume compatibility conditions for $t=0$ as in  
our main theorem.  
\end{theorem}  
\begin{proof}  
We sketch the proof which can be obtained by combining the  
proofs of \cite{Urbas Calc 1998} and of the corresponding   
Neumann boundary value problem above. The $C^0$-estimates  
follow from the maximum principle, $C^1$-estimates are  
stated in \cite{ltu}. The crucial proof of the $C^2$-estimates  
is obtained as a combination of the proof above and the proof  
of \cite{Urbas Calc 1998}, where the inequality for   
geometric and arithmetic means has to be avoided as in  
\cite{Minkowski space}. Instead we use Lemma \ref{three  
equivalences}. Higher regularity follows by standard  
theory and the considerations above give smooth convergence   
to a solution of the oblique boundary value problem for   
Hessian equations.   
\end{proof}  
  
\begin{remark}  
A similar result can be obtained in the case that   
(\ref{structure 2}) is fulfilled instead of   
$0\le F\left(D^2u_0\right)-f(x,u_0,Du_0)$.  
\par  
Again it is possible to modify the flow equation by   
introducing $\Phi$ as above.  
\par  
It is also possible to obtain the existence proof   
for the elliptic oblique boundary value problem stated in  
Theorem \ref{Appendix Theorem} by elliptic methods   
(\cite{Urbas Calc 1998}) as well as for $\beta=\nu$ by  
modifying the proof of \cite{ltu}.  
\end{remark}  
  
\subsection{Nonconvex domains}  
If $\Omega$ fails to be convex, then all the steps above work  
perfectly - provided $\varphi_z\ge c(\partial\Omega)>0$ is  
sufficiently large - besides the a priori estimates for   
$u_{\nu\nu}$ that seem to be out of reach at the moment.  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\section{Hessian quotient equations}  
\label{Curvature quotient equations section}  
  
\begin{proof}[Proof of Theorem \ref{Snk theorem}]  
We confine ourselves to the most important differences   
compared to the proof of Theorem \ref{main theorem}.  
\par  
Inequality (\ref{estimate for theta}) can be obtained  
by taking $\vartheta$ as a strictly concave function that  
vanishes on $\partial\Omega$, because $\hat g_{p_i}\equiv 0$.  
Lemma \ref{relaxed three equivalences} is needed to obtain  
(\ref{*inequality}). The original function $\vartheta$ can   
be used here.  
Lemma \ref{u beta beta lemma} follows by using Lemma   
\ref{properties of F*...}.  
 Finally, to obtain Lemma \ref{interior estimates lemma}, we  
have to use a positive lower bound for $\tr F^{ij}$. This  
bound follows from the concavity of $\hat F$ or from   
Lemma \ref{relaxed three equivalences}. Here it suffices  
to maximize   
$$w:=\log u_{\xi\xi}+\lambda\cdot|x|^2$$  
for $\lambda\gg1$.  
\end{proof}  
  
\section{Miscellaneous results}  
\begin{lemma}\label{Phi gleich Psi}  
Let $\Psi:\R2\to\R{}$ be a smooth concave function such that  
$\Psi_1>0$ or $\Psi_2<0$ and $\Psi(x,x)=0$ $\forall\,x$.  
Then there exists $\Phi:\R{}\to\R{}$ with $\Phi'>0$, $\Phi''\le 0$  
such that  
$$\Psi(x,y)=\Phi(x-y).$$  
\end{lemma}  
\begin{proof}  
The monotonicity of $\Psi$ implies that $\Psi>0$ in $\{x>y\}$,  
so the concavity of $\Psi|_{\{x-y=c\}}$, $c>0$, gives that  
$\Psi|_{\{x-y=c\}}$ is constant. We fix $\alpha>0$ and consider  
$\{(x,y):\Psi(x,y)=-\alpha\}$. Our claim follows immediately if  
we show that $\{\Psi=-\alpha\}$ is a straight line. We consider  
only the case $\Psi_1>0$. As $\Psi|_{\{x\ge y\}}$ is constant along  
$\{x-y=c\}$ we see that $\Psi_1(x,x)=:\beta>0$ is independent of $x$.  
Due to the concavity of $\Psi$ we get thus  
$$\Psi(x-\lambda,x)\le\Psi(x,x)-\Psi_1(x,x)\lambda=-\beta\lambda  
\quad\forall\,x$$  
and deduce that there exists $c>0$ such that  
$$\{\Psi=-\alpha\}\subset\{-c<x-y<0\}.$$  
Using again the concavity of $\Psi$ we see that $\{\Psi=-\alpha\}$  
is a convex curve that can be represented as  
a graph over $\{x=0\}$ due to $\Psi_1>0$. Such a curve in a strip  
as mentioned above, however, has to be a straight line.  
\end{proof}  
  
\begin{lemma}\label{Iteration} 
Let $u:\overline
\Omega\times [0,\infty)\to\R{}$ be a smooth function such that  
$$\Vert D^l u\Vert^2_{\overline\Omega}\le C_l$$  
for constants $C_l$ independent of $t$. 
If there exist positive constants $c, \tilde\lambda$ such that 
$$\Vert u\Vert^2_{\overline\Omega}\le c e^{-\tilde\lambda t},$$ 
then for any $0<\lambda<\tilde\lambda$ we can find positive constants 
$c_l$ such that 
$$\Vert D^lu\Vert^2_{\overline\Omega}\le c_le^{-\lambda t}.$$  
\end{lemma} 
\begin{proof} 
Without loss of generality we assume $\tilde\lambda=1$. 
We use interpolation inequalities of the form 
$$\Vert Dv\Vert^2\le c\Vert v\Vert\cdot(\Vert D^2v\Vert+\Vert Dv\Vert)$$ 
inductively. This 
induction gives the following sequence 
for $l,\,k\in\mathbb{N}$ 
$$a_{l,k}:=\left\{\begin{array}{ll}  
1,&l=0,\\0,&k=0,\,l>0,\\%\max\left\{\max\limits_{m<k}a_{l,m},  
\displaystyle{\frac{a_{l-1,k-1}+a_{l+1,k-1}}{2}}%\right\}
,&l>0,\,k>0,  
\end{array}\right.$$
where $a_{l,k}$ are exponents such that 
$$\Vert D^lv\Vert\le C_{l,k}e^{-a_{l,k}t}$$
for positive constants $C_{l,k}$. Here $k$ is the induction variable.
We will prove that $a_{l,k}\to1$ as $k\to\infty$.   
We have $0\le a_{l,k}\le 1$, $a_{l,k}\ge a_{l+1,k}$, and  
$a_{l,k}\le a_{l,k+1}$ for all $l,\,k\ge0$. 
Let 
$a_l:=\lim\limits_{k\to\infty}a_{l,k}$ and observe that  
$a_l=\frac{a_{l-1}+a_{l+1}}{2}$ for $l\ge1$, so we deduce  
$$a_{m+l}=a_m-l\cdot(a_m-a_{m+1}).$$  
As $1\ge a_l\ge 0$, we see that $a_l=1$ for all $l$.  
\end{proof}  
  
\end{appendix}  
  
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\end{document} 



