% O. C. Schnuerer:
% Flows Towards Reflectors 
% 07.05.2003 and Oct. 2003

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\begin{document}

\title{Flows Towards Reflectors}

\author{Oliver C. Schn\"urer}
\address{Free University Berlin,
  Department of Mathematics and Computer Science,
  Ar\-nim\-al\-lee 2--6, 14195 Berlin, Germany}
\email{schnuere@math.fu-berlin.de}

\subjclass[2000]{35K15, 35K20, 49Q10, 58J35, 78A05.}

\date{September 2001, revised May and October 2003.}

\keywords{Antenna, geometric optics, 
  parabolic Monge-Amp\`ere equation, second boundary value problem.}

\begin{abstract}
A classical problem in geometric optics is to find
surfaces that reflect light from a given light source 
such that a prescribed intensity on a target is 
realized. We introduce a flow equation for surfaces
such that they converge to solutions of this reflector
problem both for closed hypersurfaces 
and for the illumination of prescribed domains.
\end{abstract}

\maketitle
 
\markboth{OLIVER C. SCHN\"URER}
{FLOWS TOWARDS REFLECTORS}

\section{Introduction}
The classical reflector problem is to find a hypersurface 
such that light of a given intensity is reflected at this
hypersurface so that a prescribed intensity on a target
is realized. A ray of light in direction $x$ is reflected at
a hypersurface according to the reflection law to the new 
direction 
$$T(x)=x-2\langle x,\nu\rangle\nu,$$
where $\nu$ is a unit normal to the hypersurface at the point
where the ray of light is reflected. In 
\cite{GuanWang} the authors study a light source in $\R^{n+1}$,
$n\ge 2$, located at the origin emitting light in all directions
with a given smooth positive intensity function $f:S^n\to\R$,
defined on the unit sphere.
Each ray of light is reflected exactly once at a hypersurface
that is star-shaped with respect to the origin. The directions
of the reflected light correspond to points on the unit
sphere $S^n$, so the
reflection induces a new intensity function. Using elliptic
methods it is shown in \cite{GuanWang} that for any two 
intensity functions $f$ and $g$ as above there exists a 
smooth hypersurface that is starshaped with respect to the
origin such that the intensity function induced by the reflection
equals the prescribed function $g$, if the energy of the
emitted and reflected light coincide, i.\ e.\ if
\begin{equation}\label{equal energy}
\int\limits_{S^n}f=\int\limits_{S^n}g.
\end{equation} 
Moreover, the solution is unique up to dilatations 
when $T:S^n\to S^n$ is a diffeomorphism.
Using indices to denote covariant derivatives on $S^n$ with
respect to the metric $\sigma_{ij}$ induced from the standard
embedding $S^n\to\R^{n+1}$, this problem is equivalent, 
see \cite{GuanWang}, to the partial differential equation
\begin{equation}\label{u refl eqn}
\frac{\det\left(u_{ij}+\left(u-\frac{|\nabla u|^2+u^2}{2u}\right)
\sigma_{ij}\right)}{\det\left(\frac{|\nabla u|^2+u^2}{2u}\cdot
\sigma_{ij}\right)}=\frac{f(x)}{g(T(x))}
\end{equation}
for $u:S^n\to\R_+$ with positive definite matrices as arguments 
in the determinants. 
The geometric meaning of this positivity condition is explained
in \cite{GuanWang}. It means that our hypersurface lies on
one side of appropriate parabola that reflect light to one
direction.
We remark that $|\nabla u|$ is also evaluated using
the induced metric of the sphere.
The geometrical meaning of $u$ is as follows.
 For $x\in S^n\subset\R^{n+1}$, we 
define $\rho:S^n\to\R_+$ such that
$\rho(x)\cdot x$ belongs to our hypersurface. Then we have
$u(x)=\frac{1}{\rho(x)}$.
  
We give an alternative proof of the result presented above 
using a parabolic flow equation. The flow, we are going to
use, describes the deformation of reflecting hypersurfaces.
These hypersurfaces
converge finally to a stationary solution solving 
Equation (\ref{u refl eqn}). In general it is difficult to
use a parabolic flow equation to 
obtain solutions to an elliptic problem that admits
several solutions. Here it is known that any two solutions
differ by a positive multiple. As it seems easier to us to
consider a situation in which two solutions differ by an
additive constant, we introduce a new function $\phi:S^n\to
\R$ by defining $\phi(x)=\log u(x)$. 
It is easy to see that Equation
(\ref{u refl eqn}) is equivalent to
\begin{equation}\label{ell phi}
\frac{\det\left(\phi_{ij}+\phi_i\phi_j+\frac{1}{2}\left(
1-|\nabla\phi|^2\right)\sigma_{ij}\right)}{\det\left(
\frac{1}{2}\left(1+|\nabla\phi|^2\right)\sigma_{ij}
\right)}=\frac{f(x)}{g(T(x))}.
\end{equation}
% So the exterior normal to the reflector is described by
% $$\nu=\frac{1}{\sqrt{1+\frac{1}{\rho^2}|\nabla\rho|^2}}
% \left(x-\frac{\rho^i}{\rho}x_i\right)
% =\frac{1}{\sqrt{1+|\nabla\phi|^2}}\left(x+\phi^ix_i\right),$$
% and rays of light emitted in direction $x$ 
% are reflected to the direction
% $$T(x)=\frac{|\nabla\phi|^2-1}{1+|\nabla\phi|^2}x
% -\frac{2\phi^ix_i}{1+|\nabla\phi|^2}.$$
% The term $\phi^ix_i$ denotes the gradient of $\phi$ considered
% as a vector in $\R^{n+1}$.
We wish to investigate a flow that becomes stationary at
solutions of the elliptic problem and keeps the argument
of the determinant in the numerator positive definite. 
We choose the following equation
\begin{equation}\label{flow equation}
\dot\phi=\Phi\left(\log\left\{\frac{\det\left(\phi_{ij}+\phi_i\phi_j
+\frac{1}{2}\left(1-|\nabla\phi|^2\right)\sigma_{ij}\right)}
{\det\left(\frac{1}{2}\left(1+|\nabla\phi|^2\right)\sigma_{ij}
\right)}\cdot\frac{g(T(x))}{f(x)}\right\}\right)
\end{equation}
with $\Phi:\R\to\R$, $\Phi(0)=0$, $\Phi'>0$ and $\Phi''\le0$.
 For a discussion of this ansatz for the flow equation,
we refer to \cite{OSKnutAIHP}. Besides the choice
$\Phi(t)=t$, another interesting flow is obtained when
$\Phi(t)=1-e^{-\lambda t}$, $\lambda>0$, i.\ e.\ if
$$\dot\phi=1-\left(\frac{\det\left(\frac{1}{2}\left(1+|\nabla
\phi|^2\right)\sigma_{ij}\right)}{\det\left(\phi_{ij}
+\phi_i\phi_j+\frac{1}{2}\left(1-|\nabla\phi|^2\right)
\sigma_{ij}\right)}\cdot\frac{f(x)}{g(T(x))}\right)^{\lambda}.$$
We get the following
\begin{theorem}\label{main thm}
Let $f,\,g:S^n\to\R_+$ be smooth functions and
let $\phi_0:S^n\to\R$ be a smooth function such that the 
argument of the determinant in the numerator in (\ref{flow equation}) is 
positive definite. Then the evolution equation (\ref{flow equation})
with initial condition $\phi\vert_{t=0}=\phi_0$ has a solution
for all positive times, i.\ e.\ there exists a smooth function
$\phi:S^n\times[0,\infty)\to\R$ satisfying (\ref{flow
equation}). 
The function $\phi(\cdot,t)$ converges in
$C^\infty$ topology to a translating solution $\phi^\infty$
as $t\to\infty$,
i.\ e.\ there exists $v^\infty\in\R$ such that
$\phi^\infty(x,t)=\phi^\infty(x,0)+v^\infty\cdot t$.
Moreover, $v^\infty$ is determined by
\begin{equation}\label{v inf}
v^\infty=\Phi\left(\log\int\limits_{S^n}g-
\log\int\limits_{S^n}f\right),
\end{equation}
so that we get a solution to the reflector equation
(\ref{ell phi}) provided (\ref{equal energy}) holds and
the hypersurfaces induced by $\phi(\cdot,t)$ as described 
above converge to the reflector we look for as $t\to\infty$.
\end{theorem}
We remark that our parabolic approach does not only give
a constructive method to find reflectors. If (\ref{equal
energy}) is violated, a translating solution (at a fixed
time) reflects the light such that the intensity of the
reflected light equals $g$ up to a constant factor.
Note that Theorem \ref{main thm} implies the existence 
theorem in \cite{GuanWang} as $\phi_0=c\in\R$ is 
an admissible initial value.

%%%%%%%%%%%%%%%%%%%%%%%%%%
In the problem considered so far, 
the light source emits light in all
directions and light should be reflected to all directions.
Now we address a model problem of a reflector that
shall only illuminate a prescribed domain. We consider
the situation when light is emitted from a domain
$\Omega\subset\R^n\hookrightarrow\R^{n+1}$ in direction $e_{n+1}$,
where we identify $\R^n$ and $\R^n\times\{0\}$.
We assume that a hypersurface, the reflector,
 is represented as a graph over
$\Omega$ such that the light is reflected back to 
a domain $\Omega^*\subset\R^n$.

\begin{figure}[htb]
%\typeout{includes maple.ps}
\centerline{\epsfbox{maple.ps}}
\caption{Reflection at a surface} 
\label{maple}
\end{figure}

This is illustrated in Figure \ref{maple}.
There we see the upwards directed rays of light,
the reflecting surface, normals to this surface and
finally the reflected rays of light.  
For simplicity we consider the following simple model. If
the domain $\Omega$ is small compared to $\Omega^*$,
we can neglect the size of the reflector and assume that the 
reflected light is emitted from a
single point -- we take $(0,1)\in\R^n\times\R$ -- in the 
direction given by the reflection law. This problem has
applications in the design of reflectors for lamps. 

\begin{figure}[htb]
%\typeout{includes foto.ps} 
\centerline{\epsfbox{foto.ps}}
\caption{Lamp in the court yard of the Max Planck Institute
for Mathematics in the Sciences, Leipzig, Germany}
\label{foto}
\end{figure}

Figure \ref{foto}
shows a lamp in the court yard of our institute that
illuminates the ground by sending light via a reflector
to the ground. Up to now, the reflector consists of 
four triangles, so it seems desirable to improve the
shape used there. 

Using a flow ansatz similar as above we show that for
any bounded smooth strictly convex domains $\Omega$, $\Omega^*
\subset\R^n$ with $0\in\Omega^*$ and for any smooth functions
$f:\overline{\Omega}\to\R_+$, $g:\overline{\Omega^*}\to\R_+$,
there exists a hypersurface,
represented as 
$\graph\left.u\right|_{\Omega}$, such that light emitted
with intensity $f$ from $\Omega$ is reflected 
-- in our model with small $\Omega$ -- to
$\Omega^*$ and the intensity $g$ is realized provided that
$$\int\limits_\Omega f=\int\limits_{\Omega^*}g.$$
Indeed, we can solve this problem for a larger class
of domains $\Omega^*$, but to describe these domains,
it is useful to have a technical
derivation of the corresponding equations. So we give a
description of the admissible class of domains $\Omega^*$
and the formulation of the corresponding theorem 
in Section \ref{planar problems}.

In this second part we focus on the geometric description 
of the situation considered. It turns out that
we get a second boundary value problem for a
Monge-Amp\`ere equation.
This equation has been studied before in \cite{Urbas2ndCrelle}
in the elliptic setting and in a slightly different
version in \cite{OSKnutAIHP}, see also the appendix in 
\cite{OSMM}.

The purpose of the first part of this paper is to show
how solutions to closed reflector problems can be found using
parabolic techniques. The a priori estimates used here 
appear in similar form already in \cite{GuanWang}.
Note that we cannot show directly convergence to a stationary
solution. Instead, we show convergence to a translating
solution in a first step. Then we obtain that translating
solutions are indeed stationary, if \eqref{equal energy} holds.

In the second part, we introduce a model for a boundary
value problem for reflectors corresponding to a light 
source emitting parallel light. This model allows to
obtain smoothness of the reflector up to the boundary.
We also discuss another
slightly different reflector problem and indicate why
we expect that a priori estimates in this case fail to
hold. This corresponds to the fact that we can reconstruct
a reflector when we use parallel light, but cannot do so, if we
use light emitted radially from a point-like light source.

%%%%%%
It is a further issue to solve the reflector problem 
with prescribed domains
using a model that contains less simplifications.
Weak solutions of boundary value problems for 
reflectors have been considered before, see
e.\ g.\ \cite{OlikerHomepage}.
These solutions are not necessarily smooth up to the
boundary.

The paper is organized as follows. In Section
\ref{longtime existence} we prove a priori estimates and 
show that a solution to the flow equation (\ref{flow equation})
exists for all time, then we obtain convergence to a 
translating solution in Section \ref{convergence}.
In Section \ref{planar problems} we address the
problem of illuminating domains in $\R^n$ and state
the main theorem for this problem. Finally, in Section
\ref{disc sec}, we discuss the possibility to obtain
a priori estimates for a point-like light source
``outside'' a reflector.

We would like to thank J\"urgen Jost, Shing-Tung Yau, 
the Max Planck Institute for Mathematics in the Sciences,
Leipzig, Germany, the Alexander von Humboldt foundation
and Harvard University, Cambridge, MA, USA,
for discussions and support.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Longtime Existence for Closed Hypersurfaces}
\label{longtime existence}
In this section we address Theorem \ref{main thm}. It is
known that the initial value problem
(\ref{flow equation}), $\left.\phi\right|_{t=0}=\phi_0$,
admits a smooth solution for a maximal time interval
$[0,T)$. We remark that we get a similar result for
$\phi_0\in C^{2,\alpha}\left(S^n\right)$, $\alpha>0$,
with less regularity at $t=0$. To prove smooth longtime
existence, it suffices to prove that the (spatial)
$C^2$-norm of a smooth solution in a given time interval
$[0,t]$ is bounded above by $h(t)$ for any $t>0$, where
$h:\R\to\R$ is a locally bounded function. As we get
that the argument of $\Phi$ is bounded, we see that
our equation is uniformly parabolic. Thus we can 
apply Corollary 14.9 in \cite{LiebermanBook} and get
$C^{2,\alpha}$-estimates for some $\alpha>0$. Higher
regularity and uniform estimates for higher derivatives
follow from Schauder theory. 
Then it is possible to extend a solution to $[0,\infty)$
due to shorttime existence.

More precisely, we will prove uniform estimates for
$\dot\phi$, uniform oscillation estimates for $\phi$
and uniform estimates for $D\phi$ and $D^2\phi$.
Due to the $\phi$-invariance of our problem these 
estimates imply uniform estimates for all derivatives 
of $\phi$.

We will use the Einstein summation convention and lift
indices with respect to the induced metric on $S^n$.

We first bound the time derivative of $\phi$.

\begin{lemma}\label{dot phi est}
Let $\phi$ be a smooth solution of our initial value
problem. Then we have the estimate
$$\min\left\{\min\limits_{t=0}\dot\phi,0\right\}
\le\dot\phi\le\max\left\{\max\limits_{t=0}\dot\phi,0\right\}.$$
\end{lemma}
\begin{proof}
We rewrite the flow equation using 
$$\tilde f(x,\nabla\phi)=\log\det\left(\tfrac{1}{2}
\left(1+\vert\nabla\phi\vert^2
\right)\sigma_{ij}\right)-\log g(T(x))+\log f(x)$$
and
$$w_{ij}=\phi_{ij}+\phi_i\phi_j+\tfrac{1}{2}\left(1-\vert\nabla
\phi\vert^2\right)\sigma_{ij}\equiv\phi_{ij}+r_{ij}$$
here and in the following. We get
\begin{equation}\label{flow short}
\dot\phi=\Phi\left(\log\det w_{ij}-
\tilde f(x,\nabla\phi)\right).
\end{equation}
For $\dot\phi$ we obtain the evolution equation
$$\ddot\phi=\Phi'w^{ij}\dot\phi_{ij}+\Phi'w^{ij}
(2\phi_i\dot\phi_j-\phi^k\dot\phi_k\sigma_{ij})
-\Phi'\tilde f_{p_i}\dot\phi_i,$$
where the index $p_i$ indicates derivatives with respect 
to $\nabla_i\phi$ and $\left(w^{ij}\right)$ denotes the inverse of 
$(w_{ij})$. The inverse $w^{ij}$ is the only exception to our
convention to lift indices with respect to the induced metric
on $S^n$. The claimed inequality follows from the maximum
principle.
More precisely, we see that for some time interval
$(w^{ij})$ remains positive definite. During this time interval
we get the claimed inequality. Thus the argument of $\Phi$
remains uniformly bounded. From the definition of $\tilde f$,
we see that $\tilde f$ is uniformly bounded from below and 
deduce a uniform lower bound for $\log\det w_{ij}$, 
thus $(w_{ij})$ remains positive definite.
\end{proof}
Integrating this estimate, we obtain a very rough $C^0$-estimate
$$\vert\phi(x,t)\vert\le\max\vert\phi(x,0)\vert+t\cdot
\max\vert\dot\phi(x,0)\vert.$$
We need a better estimate, that prevents different parts
of the hypersurfaces from moving ``far apart" from each other. 
This is contained in the following oscillation estimate
\begin{lemma}
Let $\phi$ be a smooth solution of our initial value problem.
Then its oscillation is uniformly bounded during the flow.
\end{lemma}
\begin{proof}
We rewrite our flow equation as
\begin{equation}\label{rewr flow eqn}
\frac{\det\left(\phi_{ij}+\phi_i\phi_j+\frac{1}{2}
\left(1-\vert\nabla\phi\vert^2\right)\sigma_{ij}\right)}
{\det\left(\frac{1}{2}\left(1+\vert\nabla\phi\vert^2\right)
\sigma_{ij}\right)}=\frac{f(x)\cdot e^{\Phi^{-1}(\dot\phi)}}
{g(T(x))}.
\end{equation}
 For a fixed time $t$ we consider ${\Phi^{-1}(\dot\phi)}$
as a bounded function. Thus we can apply the $C^0$-estimates
of Section 2.1 in \cite{GuanWang} and get exactly the
claimed oscillation estimate. The $C^0$-estimate in the
cited paper is obtained for normalized surfaces, i.\ e.\
the surfaces are rescaled so that the distance of the surface 
to the origin is equal to $1$. Thus these $C^0$-estimates
correspond to oscillation estimates in our setting.
\end{proof}
The following lemma gives $C^1$-a priori estimates.
\begin{lemma}[$C^1$-estimates]
For any function $\phi\in C^2\left(S^n\right)$ with
positive definite $(w_{ij})$ (see the definition in 
the proof of Lemma \ref{dot phi est}) and bounded oscillation,
$\vert\nabla\phi\vert$ is uniformly bounded. 
\end{lemma}
\begin{proof}
The quantity
$$\tfrac{1}{2}\log\vert\nabla\phi\vert^2+\phi$$
attains its maximum somewhere on $S^n$. So we deduce there
(we multiply the covariant derivative of the quantity
above with $\phi^i$)
$$0=\frac{\phi^i\phi_{ij}\phi^j}{\vert\nabla\phi\vert^2}
+\vert\nabla\phi\vert^2.$$
As $(w_{ij})$ is positive definite, we get in the sense
of matrices 
$$\phi_{ij}\ge-\phi_i\phi_j-\tfrac{1}{2}\left(1-\vert\nabla
\phi\vert^2\right)\sigma_{ij}$$
and deduce at the maximum point
$$1\ge\vert\nabla\phi\vert^2.$$
Since the oscillation of $\phi$ is bounded, we get a uniform
bound for $\vert\nabla\phi\vert$ everywhere on $S^n$.
\end{proof}

Before we estimate the second covariant derivatives of 
$\phi$, we recall formulae for interchanging the 
order of covariant differentiation for functions on $S^n$
\begin{eqnarray*}
\phi_{ijk}&=&\phi_{kij}+\phi_j\sigma_{ik}
-\phi_k\sigma_{ij},\\
\phi_{ijkl}&=&\phi_{klij}+2\phi_{ij}\sigma_{kl}
-2\phi_{kl}\sigma_{ij}+\phi_{kj}\sigma_{il}-\phi_{il}
\sigma_{kj}.
\end{eqnarray*}
 
\begin{lemma}[$C^2$-estimates]
The second covariant spatial derivatives of $\phi$ remain uniformly 
bounded during the flow.
\end{lemma}
\begin{proof}
We use the maximum principle for $\sigma^{ij}w_{ij}$ and compute its
evolution equation. We will rewrite 
$$\dot w_{ij}-\Phi'w^{kl}w_{ijkl}$$
using terms we are able to control. The last two indices
of $w_{ijkl}$ denote covariant derivatives on $S^n$.
We use the definition 
\begin{equation}\label{w def}
w_{ij}=\phi_{ij}+r_{ij},
\end{equation}
differentiate this equation and use it to substitute 
$\dot w_{ij}$ and $w_{ijkl}$. 
Derivatives of $r_{ij}$ with respect to $x$ vanish.
Next, we differentiate
the flow equation (\ref{flow short}) twice in
spatial directions and replace $\dot\phi_{ij}$ using
this equation
\begin{align*}
\dot w_{ij}-\Phi'w^{kl}w_{ijkl}=&
\Phi'w^{kl}w_{klij}-\Phi'w^{kl}\phi_{ijkl}\\
&-\Phi'w^{kr}w^{ls}w_{kli}w_{rsj}
+r_{ijp_r}\left(\dot\phi_r-\Phi'w^{kl}\phi_{rkl}\right)\\
&-\Phi'w^{kl}r_{ijp_rp_s}\phi_{rk}\phi_{sl}
-\Phi'D_jD_i\tilde f\\
&+\Phi''\left(w^{ab}w_{abi}-D_i\tilde f\right)
\left(w^{cd}w_{cdj}-D_j\tilde f\right).
\end{align*}
The notation $D_\cdot$ indicates that the chain rule has not yet
been applied to the respective terms. 
We rewrite $w_{ijkl}$ in terms of derivatives
of $\phi$ and $r_{ij}$ and interchange derivatives of
$\phi_{ijkl}$. So the terms containing fourth derivatives 
of $\phi$ drop out. 
The quantity $r_{kl}$ depends on $(x,\nabla\phi)$,
but its covariant derivatives with respect to the $x$ 
variable vanish.
\begin{align*}
\dot w_{ij}-\Phi'w^{kl}w_{ijkl}=&
\Phi'w^{kl}(2\phi_{kl}\sigma_{ij}-2\phi_{ij}\sigma_{kl}
+\phi_{il}\sigma_{kj}-\phi_{kj}\sigma_{il})\\
&{}+r_{ijp_r}\dot\phi_r-\Phi'w^{kl}r_{ijp_r}\phi_{rkl}
+\Phi'w^{kl}r_{klp_r}\phi_{rij}\\
&{}-\Phi'w^{kl}r_{ijp_rp_s}\phi_{rk}\phi_{sl}+\Phi'w^{kl}r_{klp_rp_s}
\phi_{ri}\phi_{sj}\\
&{}-\Phi'w^{kr}w^{ls}w_{kli}w_{rsj}-\Phi'D_jD_i\tilde f\\
&{}+\Phi''\left(w^{ab}w_{abi}-D_i\tilde f\right)
\left(w^{cd}w_{cdj}-D_j\tilde f\right).
\end{align*}
We interchange third derivatives of $\phi$, use
\eqref{flow short} and \eqref{w def}
\begin{align*}
r_{ijp_r}\dot\phi_r&-\Phi'w^{kl}r_{ijp_r}\phi_{rkl}
+\Phi'w^{kl}r_{klp_r}\phi_{rij}\\
=&r_{ijp_r}\left(\dot\phi_r-\Phi'w^{kl}\phi_{klr}\right)
+r_{ijp_r}\Phi'w^{kl}(\phi_l\sigma_{kr}-\phi_r\sigma_{kl})\\
&+\Phi'w^{kl}r_{klp_r}(\phi_{ijr}-\phi_j\sigma_{ir}
+\phi_r\sigma_{ij})\\
=&r_{ijp_r}\left(\Phi'w^{kl}r_{klp_s}\phi_{sr}
-\Phi'D_r\tilde f\right)
+r_{ijp_r}\Phi'w^{kl}(\phi_l\sigma_{kr}-\phi_r\sigma_{kl})\\
&+\Phi'w^{kl}r_{klp_r}(w_{ijr}-r_{ijp_s}\phi_{sr}
+\phi_r\sigma_{ij}-\phi_j\sigma_{ir}).
\end{align*}
So we get the evolution equation
\begin{align}
\begin{split}\label{evol 1}
\dot w_{ij}-\Phi'w^{kl}w_{ijkl}=&
\Phi'w^{kl}(2\phi_{kl}\sigma_{ij}-2\phi_{ij}\sigma_{kl}
+\phi_{il}\sigma_{kj}-\phi_{kj}\sigma_{il})\\
&{}+\Phi'r_{ijp_r}\left(
w^{kl}(r_{klp_s}\phi_{sr}
-\phi_r\sigma_{kl}+\phi_l\sigma_{kr})-D_r\tilde f\right)\\
&{}-\Phi'w^{kl}r_{ijp_rp_s}\phi_{rk}\phi_{sl}+\Phi'w^{kl}r_{klp_rp_s}
\phi_{ri}\phi_{sj}\\
&{}+\Phi'w^{kl}r_{klp_r}(w_{ijr}-r_{ijp_s}\phi_{sr}
+\phi_r\sigma_{ij}-\phi_j\sigma_{ir})\\
&{}-\Phi'w^{kr}w^{ls}w_{kli}w_{rsj}-\Phi'D_jD_i\tilde f\\
&{}+\Phi''\left(w^{ab}w_{abi}-D_i\tilde f\right)
\left(w^{cd}w_{cdj}-D_j\tilde f\right).
\end{split}
\end{align}
Directly from the definitions of $r_{kl}$ and $w_{kl}$ we obtain
\begin{align}\label{crucial term}
\begin{split}
\lefteqn{-w^{kl}r_{ijp_rp_s}\phi_{rk}\phi_{sl}+w^{kl}r_{klp_rp_s}
\phi_{ri}\phi_{sj}}\\
=&w^{kl}\phi_{lr}\sigma^{rs}\phi_{sk}\cdot\sigma_{ij}
-w^{kl}\sigma_{kl}\cdot w_{ir}\sigma^{rs}w_{sj}\\
&{}+w^{kl}\sigma_{kl}\cdot
\left(w_{ir}\sigma^{rs}r_{sj}+w_{jr}\sigma^{rs}r_{si}\right)
-w^{kl}\sigma_{kl}\cdot r_{ir}\sigma^{rs}r_{sj}.
\end{split}
\end{align}
The term $-w^{kl}\sigma_{kl}\cdot w_{ir}\sigma^{rs}w_{sj}$
will be very useful for further estimates. We remark 
that the right-hand side of (\ref{evol 1}) is a tensor
with indices $i$ and $j$ and this is also true for both
terms on the left-hand side. 
Let us note that $r_{ij}=w_{ij}-\phi_{ij}$ is bounded.
We denote $w^{kl}\sigma_{kl}$ by $\tr w^{kl}$ and obtain
$$\left\vert w^{kl}\phi_{lj}\right\vert=
\left\vert w^{kl}w_{lj}-w^{kl}(w_{lj}-\phi_{lj})\right\vert
\le 1+\left\vert w^{kl}r_{lj}\right\vert
\le c\cdot\left(1+\tr w^{kl}\right).$$ 
The matrix $w_{ij}$ is positive definite, so it suffices
to estimate $W:=\sigma^{ij}w_{ij}$ from above in order
to prove boundedness of the second derivatives of $\phi$.
The metric of the sphere, $\sigma_{ij}$, is parallel with
respect to covariant differentiation on $S^n$ and 
time-independent. So it is easy to combine 
\eqref{evol 1} and \eqref{crucial term} to obtain the
evolution equation for $W$. We use concavity of $\Phi$,
boundedness of $\Phi'$ and interchange third derivatives
of $\phi$ in $D_jD_i\tilde f$
\begin{align*}
\dot W-\Phi'w^{kl}W_{kl}\le&\Phi'w^{kl}\phi_{lr}\sigma^{rs}
\phi_{sk}\cdot n-\Phi'w_{ir}\sigma^{rs}w_{sj}\sigma^{ij}
\cdot\tr w^{kl}\\
&+\Phi'w^{kl}r_{klp_r}W_r-\Phi'\tilde f_{p_r}W_r\\
&+c\cdot\left(1+\tr w^{kl}+\left\lvert D^2\phi\right\rvert
\cdot\tr w^{kl}+\left\lvert D^2\phi\right\rvert^2\right).
\end{align*}
Using $\phi_{ij}=w_{ij}-r_{ij}$ once again, the first term
on the right-hand side can be absorbed in the error term and
$\left\lvert D^2\phi\right\rvert$ in the error term can be
replaced by $W$. Assume now that there exists
$(x_0,t_0)\in S^n\times(0,T)$ such that 
$W(x_0,t_0)\ge W(x,t)$ for $(x,t)\in S^n\times[0,t_0]$.
Then the parabolic maximum principle implies that
$$0\le\dot W-\Phi'w^{kl}W_{kl},\qquad W_r=0.$$
For positive $a_i$, we have $\tfrac1n\sum\limits_{i,j=1}^n
a_ia_j\le\sum\limits_{i=1}^na_i^2.$ We employ this 
inequality to the eigenvalues of $w_{ij}$ with respect
to $\sigma_{ij}$ and deduce at $(x_0,t_0)$ that
$$\tfrac1n\Phi'W^2\cdot\tr w^{kl}\le c\cdot
\left(1+\tr w^{kl}+W\cdot\tr w^{kl}+W^2\right).$$

We use that $\Phi'$ is bounded from below
by a positive constant. Moreover, as $\log\det w_{kl}$ is
bounded, we see that $W\to\infty$ forces
$\tr w^{kl}\to\infty$. We deduce that $W$ is bounded
at $(x_0,t_0)$ and get a time-independent bound for $\left\vert
D^2\phi\right\vert$ as long as a smooth solution of
(\ref{flow equation}) exists.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convergence for Closed Hypersurfaces}
\label{convergence}
Here we complete the proof of Theorem \ref{main thm}.
The method used in \cite{OSMM} to obtain a 
translating solution also applies to the case of
closed hypersurfaces. Indeed, the proof is a bit
simpler in the closed case. For convenience of the
reader, we sketch the argument given there.
Part of the argument is due to Huisken \cite{AltschulerWu}.

Fix $t_0>0$. We consider
$$w(x,t):=\phi(x,t+t_0)-\phi(x,t).$$
Using the mean value theorem, we see that $w$ satisfies
a parabolic flow equation of the form
\begin{equation}\label{interpol eqn}
\dot w=a^{ij}w_{ij}+b^iw_i.
\end{equation}
The strong maximum principle shows that the oscillation
of $w$ is strictly decreasing during the flow or $w$ is
constant. We wish to show that
the oscillation does not tend to $\epsilon>0$. Otherwise
we consider for $x_0\in S^n$ fixed and $t_n\to\infty$
\begin{equation}\label{seq inf}
\phi(x,t+t_n)-\phi(x_0,t_n)\quad\mbox{and}\quad
\phi(x,t+t_0+t_n)-\phi(x_0,t_0+t_n).
\end{equation}
Due to our a priori estimates we can find a subsequence
such that the expressions above converge locally uniformly
(in $S^n\times(-\infty,\infty)$) in any $C^k$-norm to a solution of our flow 
equation for all time. 
It is easy to see that the difference of the
limits solves a parabolic equation similar to (\ref{interpol 
eqn}) and has
constant oscillation $\epsilon>0$. This contradicts 
the strong maximum principle. As the oscillation of $w$
tends to zero and $w$ satisfies a parabolic equation
of the form (\ref{interpol eqn}),
we see that $w$ tends to some constant as $t\to\infty$.
Considering sequences similar to (\ref{seq inf}) we
obtain a solution $\phi^*$ for all time. One checks that 
\begin{equation}\label{disc trans}
\phi^*(x,t+t_0)-\phi^*(x,t)=const..
\end{equation}
Next, we take an appropriate number,
e.\ g.\ $t_0\cdot\sqrt{2}$, instead of $t_0$ and 
start with the solution $\phi^*$ obtained. 
Our procedure gives a solution that satisfies (\ref{disc trans})
(with a different constant)
also for $t_0\cdot\sqrt{2}$ instead of $t_0$, i.\ e.\ 
we obtain a translating solution. 
Now we compare our original solution with the translating
solution. As above, we get that the oscillation of the difference
tends to zero. Smooth convergence to a translating solution
is then obtained by using interpolation inequalities.

Thus $\phi$ 
converges smoothly to a translating solution $\phi^\infty$
of (\ref{flow equation}) for $t\to\infty$.
To check that the velocity $v^\infty$ is as claimed
in (\ref{v inf}), we use the flow equation (\ref{flow
equation}) in the form (\ref{rewr flow eqn}) for the
translating solution $\phi^\infty$. We consider
this equation as an elliptic equation and obtain
from the conservation of energy and the derivation
of the elliptic reflector equation, see the appendix
in \cite{GuanWang}, 
$$\int\limits_{S^n}e^{\Phi^{-1}(v^\infty)}f=
\int\limits_{S^n}g.$$
This implies (\ref{v inf}), as $v^\infty$ is a constant,
and completes the proof of Theorem \ref{main thm}.

We wish to remark, that we can enclose our initial
function $\phi$ from above and from below by the
translating solutions obtained. Due to the maximum
principle, these translating solutions act as barriers
and show that our solutions stay at a finite distance
to a translating solution.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Illuminating Prescribed Domains}
\label{planar problems}

We start with a derivation of the equation 
fulfilled by solutions. Therefore we
follow light that moves upwards from
$(x,0)\in\R^n\times\R$, $x\in\Omega$, in
direction $(0,1)$. The reflector is described
as $\graph\left.u\right|_{\Omega}$. A unit normal
to this hypersurface is given by
$$\nu=\frac{(-Du,1)}{\sqrt{1+|Du|^2}}.$$
The direction of the reflected light is obtained
as a function of $x$ as follows
$$x\mapsto(0,1)-2\langle(0,1),\nu\rangle\nu=
\frac{\left(2Du,-1+|Du|^2\right)}{1+|Du|^2}.$$
Due to our hypotheses that in the simplified
model the reflected rays of light start at 
$(0,1)$, we see that this ray of light meets
the ``ground", i.\ e.\ the hyperplane $\R^n\times\{0\}$, at 
$\frac{2Du}{1-|Du|^2}.$
Thus we get a map $T:\Omega\to\Omega^*$ such that
light from $x$ is reflected to $T(x)$, in a formula
$$T(x)=\frac{2Du}{1-|Du|^2}.$$
It is easy to see, that $T$ is a diffeomorphism 
onto its image for
a smooth strictly convex function $u$ with $|Du|<1$; we will 
assume this in the following.

Next we derive the equation to be fulfilled by $u$.
We assume that $u$ is a solution to our reflector
problem. From the conservation of energy and the
transformation formula for integrals, we get 
for open domains $E\subset\Omega$
$$\int\limits_{T(E)}g(y)dy=\int\limits_Ef(x)dx
=\int\limits_{T(E)}f(x)\cdot\frac{1}{\det T_{ij}}dy,$$
where $T_{ij}$ denotes the derivative of the $i$-th
component of $T$ in direction $j$ and $y=T(x)$.
%We assume that all the functions involved are smooth.
Thus we obtain the elliptic equation for the
reflecting hypersurface
$$\det T_{ij}=\frac{f(x)}{g(T(x))}.$$
More explicitly, we use the Einstein summation convention and get
$$T_{ij}=\frac{\partial T_i}{\partial x^j}=
\frac{2}{\left(1-|Du|^2\right)^2}\left(u_{ij}\left(1-|Du|^2\right)
+2u_iu_l\delta^{lk}u_{kj}\right).$$
 For the evaluation of the determinant of $T_{ij}$, we 
may assume without loss of generality, that we have chosen
coordinates such that
$\langle Du,e_1\rangle=|Du|$.
$(T_{ij})$ is then given by
$$\frac{2}{\left(1-|Du|^2\right)^2}
\left(\begin{array}{cccc}
u_{11}\left(1+|Du|^2\right) & u_{12}\left(1-|Du|^2\right) & \cdots
& u_{1n}\left(1-|Du|^2\right)\\
u_{12}\left(1+|Du|^2\right) & u_{22}\left(1-|Du|^2\right) & \cdots
& u_{2n}\left(1-|Du|^2\right)\\
\vdots & \vdots & \ddots & \vdots\\
u_{1n}\left(1+|Du|^2\right) & u_{2n}\left(1-|Du|^2\right) & \cdots
& u_{nn}\left(1-|Du|^2\right)\\
\end{array}\right),$$
so we see immediately that
$$\det T_{ij}=
2^n\cdot\left(1-|Du|^2\right)^{-n-1}\cdot\left(1+|Du|^2\right)
\cdot\det D^2u$$
and the reflector equation 
$$\det D^2u=\frac{f(x)}{g(T(x))}\cdot 2^{-n}\cdot
\frac{\left(1-|Du|^2\right)^{n+1}}{1+|Du|^2}$$ 
follows. In our approach, we consider the flow equation
\begin{equation}\label{hess eqn}
\dot u=\Phi\left(
\log\det D^2u-\log\left(\frac{f(x)}{g(T(x))}\cdot 2^{-n}\cdot
\frac{\left(1-|Du|^2\right)^{n+1}}{1+|Du|^2}\right)
\right)
\end{equation}
with $\Phi$ as in (\ref{flow equation}).
The inverse map to 
$$Du\mapsto\frac{2Du}{1-|Du|^2}$$
is given by
\begin{equation}\label{intro tau}
\tau:y\mapsto\frac{y}{|y|^2}\left(\sqrt{1+|y|^2}-1\right).
\end{equation}
From the Taylor expansion of the square root 
at $y=0$, we see that
$\tau$ extends smoothly to $y=0$. 
The map $\tau$ is a diffeomorphism
onto its image, so we can rewrite the boundary condition
$T(\Omega)=\Omega^*$ as $Du(\Omega)=\tau(\Omega^*)$.
Directly from the estimates in \cite{OSKnutAIHP} and the
appendix in \cite{OSMM} we obtain
\begin{theorem}
Let $\Omega$, $\Omega^*\subset\R^n$ be smooth bounded
domains such that $\Omega$ and $\tau(\Omega^*)$ are strictly
convex domains where $\tau$ is the diffeomorphism introduced
in (\ref{intro tau}).
Let $u_0:\ol{\Omega}\to\R$ be a smooth strictly convex
function such that $Du_0(\Omega)=\tau(\Omega^*)$. Then there
exists a smooth solution $u:\ol{\Omega}\times(0,\infty)$
to Equation (\ref{hess eqn}) --
with $u(\cdot,t)\to u_0$ in $C^2\left(\ol\Omega\right)$ 
as $t\downarrow0$ --
such that $Du(\Omega)=\tau(\Omega^*)$ or equivalently
$T(\Omega)=\Omega^*$ ($T$ is evaluated using
$u(\cdot,t)$). The function $u(\cdot,t)$ converges 
in $C^\infty\left(\ol\Omega\right)$ topology
to a translating
solution of (\ref{hess eqn}) that moves with speed
$$\Phi\left(\log\int\limits_{\Omega^*}g-\log\int\limits_\Omega f
\right).$$ 
\end{theorem}
\begin{proof}
The existence and convergence to a translating solution follows
from \cite[Theorem A.1]{OSMM} where we use 
estimates from \cite{OSKnutAIHP}. Using the conservation
of energy and the transformation formula for integrals as in the 
derivation of the reflector equation above, implies for a
translating solution with velocity $v^\infty$
$$\int\limits_{\Omega^*}g(y)dy=\int\limits_{\Omega}
e^{\Phi^{-1}\left(v^\infty\right)}f(x)dx.$$
Thus we get the formula for $v^\infty$.
\end{proof}  

We remark that the maximum principle shows uniqueness
of translating solutions up to additive constants. Once again,
our solution becomes stationary provided that the total amount
of energy emitted and prescribed on the ground coincide. 

At a first glance, the convexity condition for $\tau\left(
\Omega^*\right)$ seems artificially. As it turns out, however,
that our problem corresponds to a second boundary value
problem for a Monge-Amp\`ere equation, which can be
solved -- at least at the moment -- in general only for
strictly convex domains, we see that our condition for
$\Omega^*$ is indeed natural. We show in Lemma \ref{exa lem}
that the convexity condition for $\tau\left(\Omega^*\right)$
is fulfilled for a large class of domains.

It remains to prove the assertion of the introduction that
this illumination problem can be solved for strictly convex
domains $\Omega^*$ that contain the origin, i.\ e.\ it 
suffices to prove
\begin{lemma}\label{exa lem}
Let $\Omega^*\subset\R^n$ be a convex open set, $0\in\Omega^*$. 
For $\tau$ as in (\ref{intro tau}), $\tau(\Omega^*)$ 
is strictly convex. 
\end{lemma}
\begin{proof}
As $\tau$ maps each point $x\in\R^n$ to a point $\lambda\cdot x$
where $\lambda=\lambda(|x|)$, we see that it suffices to prove
this lemma for $\Omega^*\subset\R^2$. Moreover, as $|x|\mapsto
|\tau(x)|$ is a strictly monotone increasing function, we only 
have to check that $\tau$ maps half-planes containing the origin
to strictly convex sets. Due to the rotational symmetry, it 
suffices to show that horizontal lines lying ``above" the origin 
are mapped to graphs
over part of the horizontal axis, $\graph u$, such that $u$ is 
a strictly concave positive function. 
More precisely, we fix $a>0$ and
consider the horizontal line in $\R^2$ parameterized by
$\R\ni t\mapsto(t,a)$. The diffeomorphism $\tau$ maps this
line to 
$$t\mapsto(t,a)\cdot\frac{\sqrt{1+a^2+t^2}-1}{a^2+t^2}
\equiv(t,a)\cdot g(t)\equiv(x(t),y(t)).$$
Direct calculations show that
$$\fracp{x}{t}=\frac{\left(t^2-a^2\right)\cdot
\left(\sqrt{1+a^2+t^2}-1\right)+a^2\cdot\left(a^2+t^2\right)}
{\left(a^2+t^2\right)^2\sqrt{1+a^2+t^2}}>0.$$
Thus we can use $x$ to parameterize the image.
We use the chain rule and obtain
\begin{align*}
\fracp{^2y}{x^2}=&\fracp{^2y}{t^2}\left(\fracp{t}{x}\right)^2
+\fracp yt\fracp{^2t}{x^2}\\
=&\left(\fracp{^2y}{t^2}-\fracp yt\fracp{^2x}{t^2}\fracp tx
\right)\cdot\left(\fracp{t}{x}\right)^2.
\end{align*}
Thus it suffices to show that
\begin{equation}\label{ggg}
2\left(g'(t)\right)^2>g(t)\cdot g''(t).
\end{equation}
We note that 
$$g(t)=\frac{1}{1+f},\quad\mbox{where~}f(t):=\sqrt{1+a^2+t^2},
\quad\mbox{so~}f'=\frac{t}{f}.$$
Now it is easy to obtain (\ref{ggg}) by direct calculation.
Thus our lemma follows.
% Direct calculations show that this is equivalent to
% $$4+3a^2+3t^2<\sqrt{1+a^2+t^2}\cdot
% \left(4+a^2+t^2\right).$$
% The left-hand side of this inequality is positive, so 
% we can take squares on each side and see that our lemma
% follows.
\end{proof}

\section{Radial Light from Outside}
\label{disc sec}
In Sections \ref{longtime existence} and \ref{convergence},
we have considered closed reflectors. We assumed that
$\phi=-\log\rho$ is such that 
$$\phi_{ij}+\phi_i\phi_j+\tfrac12\left(1-\vert\nabla\phi\vert^2
\right)\sigma_{ij}$$
is positive definite. This admissibility means geometrically 
that for every point on the reflector, there exists a 
paraboloid with focus at the origin that touches the
hypersurface from outside at the given point \cite{GuanWang}.
These paraboloids play a special role for this problem as they
reflect the emitted light such that it becomes parallel.

Another type of admissibility is obtained when those paraboloids
touch so that they separate the hypersurface from the origin.
We refer to it as exterior reflector problem. If our light
source lies outside a given ball, the ``illuminated part'' 
of its boundary is admissible for the exterior reflector
problem. Let $x\in S^n$ and $\rho:\Omega\to\R^n$, $\Omega
\subset S^n$, then the hypersurface given by
$\{\rho(x)\cdot x:x\in\Omega\}$ is admissible, if for 
$\phi=\log\rho$
$$\phi_{ij}-\phi_i\phi_j-\tfrac12
\left(1-\vert\nabla\phi\vert^2\right)\sigma_{ij}$$
is positive definite. The corresponding reflector equation is 
\begin{equation}\label{ext}
\frac{\det\left(\phi_{ij}-\phi_i\phi_j-\frac12\left(1-\vert
\nabla\phi\vert^2\right)\sigma_{ij}\right)}{\det\left(\frac12
\left(1+\vert\nabla\phi\vert^2\right)\sigma_{ij}\right)}
=\frac{f(x)}{g(T(x))}.
\end{equation}
Suppose that we want to prove a priori estimates for this
equation subject to a suitable boundary condition. 
Assume that we
have $C^1$-estimates in a ball on $S^n$ and $C^2$-estimates
at its boundary. Even in this situation, it seems impossible
to us to prove interior $C^2$-estimates. The reason for this is,
that the term $\frac12\vert\nabla\phi\vert^2\sigma_{ij}$
in the numerator in \eqref{ext} gives rise to inestimable
terms. They correspond to $-w^{kl}\sigma_{kl}\cdot
w_{ir}\sigma^{rs}w_{sj}$ in \eqref{crucial term} but
appear here with a different sign. Numerical experiments
confirm that we should not expect to get these estimates. 
Our simulation produces smooth
solutions to \eqref{hess eqn} but with gradient terms as in
the exterior reflector problem, we obtain interior 
singularities for different initial values.

Constructing explicit counterexamples seems hard, as
\cite{OlikerRad} shows, that there are no axially symmetric
interior counterexamples to $C^2$-regularity. Moreover,
it is proved in \cite{OlikerNearRad} that the corresponding
linearized operator is formally self-adjoint and an 
inverse function theorem argument yields almost axially
symmetric solutions. The counterexamples by Pogorelov
\cite{PogorelovBook} and Heinz-Levy \cite{SchulzBook}
have singularities that extend to the boundary.

It is already mentioned in \cite{WangInverse}, that it
might be difficult to prove these a priori estimates.
In \cite{ViaclovskyCAG}, a similar equation appears for
conformally deformed metrics, for which the product of
the eigenvalues of the Schouten tensor is prescribed. A priori
estimates up to $C^1$ are proved, but an estimate for
second derivatives could not be obtained in the situation
corresponding to \eqref{ext}. We wish to
emphasize, that $C^2$-estimates can be obtained, when there
is a $-\frac12\vert\nabla\phi\vert^2\sigma_{ij}$ term in
the determinant, see e.\ g.\ Section \ref{longtime existence},
even local interior a priori estimates are true 
\cite{GuanWang,GuanLocalSchouten}, whereas a term
$+\frac12\vert\nabla\phi\vert^2\sigma_{ij}$ seems
to make $C^2$-estimates impossible. 

The situation considered in Section \ref{planar problems}
is geometrically similar to the exterior reflector problem,
where $C^2$-estimates fail to hold. This means that (in the
appropriate model) surfaces reflecting parallel light can
be reconstructed, but this seems in general impossible for
light emitted from a point-like light source.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliographystyle{/home/napier/schnuere/os/tools/amsone}
%\bibliography{/home/napier/schnuere/os/tools/biblio}
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  \def\cprime{$'$}
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\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\begin{thebibliography}{10}
\newcommand{\hpeprint}[1]{%
  \href{http://arXiv.org/abs/#1}{\texttt{#1}}}%
\newcommand{\hpspires}[1]{%
  \dhref{http://www.slac.stanford.edu/spires/find/hep/www?#1}{\ (spires)}}%

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\end{document}

\bibliographystyle{amsplain}
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\end{document} 




