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\markboth{\hfil Log-concavity in some parabolic problems \hfil EJDE--1999/19}
{EJDE--1999/19\hfil Antonio Greco \& Bernd Kawohl \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 1999}(1999), No.~19, pp. 1--12. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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Log-concavity in some parabolic problems
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35B50, 35K55, 35K65, 35B05.
\hfil\break\indent
{\em Key words and phrases:}
Concavity maximum principle, Log-concavity, Parabolic equation.
\hfil\break\indent
\copyright 1999 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted March 23, 1999. Published May 28, 1999.\hfil\break\indent
The first author was supported by DAAD and by Progetto Finanziato
dall'Universit\`a
\hfil\break\indent
di Cagliari.
} }
\date{}
%
\author{Antonio Greco \& Bernd Kawohl}
\maketitle
\begin{abstract}
We improve a concavity maximum principle for parabolic equations of the
second order, which was initially established by Korevaar, and then we use
this result to investigate some boundary value problems. In particular, we
find structural conditions on the equation, and suitable conditions on the
domain of the problem and on the boundary data, that suffice to yield
spatial log-concavity of the (positive) solution. Examples and
applications are provided, and some unsolved problems are pointed out. We
also survey some classical as well as recent contributions to the subject.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newcommand{\eq}[1]{{\rm(\ref{#1})}}
\newcommand{\en}[1]{{\rm\ref{#1}}}
\section{Introduction}
In the last two decades we have seen many new results on qualitative
properties of solutions to elliptic and parabolic problems. One of the
issues that have been investigated is how the shape of the underlying
domain influences the shape of the solution. There is a vast literature
addressing symmetry questions, and there are also several papers
investigating convexity properties of solutions.
In the early 80's, R.~Finn posed the question whether capillary surfaces on
convex domains are convex, and he gave this problem to N.~J.~Korevaar as a
Ph.D.\ thesis. Korevaar wrote two papers on convexity that have become
classical (\cite{Ko}, \cite{Koo}), and he remarked that:
\smallskip
\begin{enumerate}\itemsep=\smallskipamount
\item Convexity of the domain alone, usually does not induce convexity of
the solution. A typical condition to be added to the problem in order
to obtain convexity of the solution is that {\it the contact angle at
the boundary must be zero}.
\item Even if the differential operator is very simple, and it is
not the minimal surface operator, there are counterexamples.
For instance, the negative first eigenfunction $-u_1$ of the
Laplace
operator, with homogeneous Dirichlet boundary conditions on the disc,
is not convex. However, it turns out that $-\log u_1$ is convex
(\cite{Koo}, Remark 2.7; see also \cite{Kaa}, Remark 3.4).
\item It would be interesting to study convexity {\it of the level
sets} of the solution, instead of convexity of the solution itself.
\end{enumerate}
\smallskip
In this paper, in order to study convexity of level sets of a positive
function $u$ we study the convexity of the auxiliary function $v:=-\log
u$. This auxiliary function is chosen because it leads to a particularly
useful structure of the transformed equation. Our main results are Theorem
\en{ConcavityMP} (a concavity maximum principle) and Theorem \en{general}
(its application to initial boundary value problems). For a review of other
techniques we refer to \cite{Kaa} (see also \cite{CF}, \cite{GP} and
\cite{Kaaa}). For an approach using viscosity solutions see \cite{ALL}.
\section{Parabolic maximum principle}
Parabolic inequalities, like \eq{PI} below, satisfy a nondegeneracy
condition in the sense that the coefficient of $u_t$
is nonzero. This property, which of course does not depend on
the ellipticity constants of the matrix $(a^{ij})$, allows the following
(well-known) maximum principle to hold.
\begin{theorem}[Weak maximum principle for parabolic equations]
\hfill\break
Let $w\in C^2({\cal G})$ be a classical solution of
%
\begin{equation}\label{PI}
w_t\le a^{ij}(x,t)\,w_{ij}+b^i(x,t)\,w_i-c(x,t)\,w
\end{equation}
%
in the set\/ ${\cal G}:=\Omega\times(0,T\,]$, $\Omega$ a bounded domain in
${\mathbb R}^N$, $T>0$. The coefficients $a^{ij}$, $b^i$ and $c$ are supposed to be
real valued functions on ${\cal G}$ satisfying $(a^{ij})\ge 0$ and\/
$\inf_{(x,t)\in{\cal G}} c(x,t)>-\infty$. Inequality\/ \eq{PI} must hold pointwise.
\\
If\/ $\limsup w(x,t)\le 0$ as $(x,t)$ approaches the {\em parabolic
boundary} of\/ ${\cal G}$ (i.e., the set $\Omega\times\{0\}
\cup\partial\Omega\times[\,0,T\,]$), then we have $w\le 0$ in all of\/
${\cal G}$.
\end{theorem}
\paragraph{Proof.}
Choose $m<\inf c$ and consider $\tilde w:=w\,e^{mt}$. By substitution into
\eq{PI} we have
$$
\tilde w_t\le a^{ij}\,\tilde w_{ij}+b^i\,\tilde w_i-\tilde c\,\tilde w,
$$
where $\tilde c=c-m>0$. Suppose, contrary to the claim, that $w$ becomes
somewhere positive. The function $\tilde w$ would still be positive there,
and a maximizing sequence would converge to some point $(x_0,t_0)$ outside
the parabolic boundary, due to the boundary behaviour of $w$. Since $\tilde
w$ is smooth at $(x_0,t_0)$, a standard computation contradicts the
inequality above and the claim follows.
\hfill$\diamondsuit$
\paragraph{Remark.}
If the assumptions of the theorem are strengthened, namely if in addition
we suppose that $\lambda\,|\xi|^2\le
a^{ij}(x,t)\,\xi_i\,\xi_j\le\Lambda\,|\xi|^2$ for all $(x,t)\in{\cal G}$,
$\xi\in{\mathbb R}^N$, with suitable $\lambda,\Lambda>0$ ({\it uniform
parabolicity}), and $\sup_{(x,t)\in{\cal G}}|b^i(x,t)|<+\infty$ for
$i=1,\dots,N$, then the {\it strong maximum principle} holds, i.e., if
$u=0$ at some point outside the parabolic boundary then $u$ is identically
zero (see, for instance, \cite{PW} or \cite{RR}).
\noindent The weak maximum principle, instead, has the
following noteworthy properties:
\begin{enumerate}\itemsep=0pt
\item No nondegeneracy condition is imposed to the matrix $(a^{ij})$.
\item No boundedness of the coefficients $a^{ij}$ and $b^i$ is assumed.
\end{enumerate}
Note, further, that the coefficients in \eq{PI} need not be smooth.
\begin{corollary}[Weak maximum principle on an infinite cylinder]
\label{MP}
\hfill\break
Let $w\in C^2({\cal S})$ be a classical solution of\/ \eq{PI} in the infinite
cylinder\/ ${\cal S}:=\Omega\times{\mathbb R}^+$, $\Omega$ a bounded domain in
${\mathbb R}^N$. Suppose that $(a^{ij})\ge 0$, and that\\
$\inf_{(x,t)\in\Omega\times(0,T)}
c(x,t)>-\infty$ for every (finite) $T>0$.
\\
If\/ $\limsup w(x,t)\le 0$ as $(x,t)$ approaches $\partial{\cal S}$, then we have
$w\le 0$ in all of ${\cal S}$.
\end{corollary}
\paragraph{Proof.}
If we assume $w>0$ at a certain $(x,T)\in{\cal S}$ then we reach a contradiction
to the preceding theorem.
\hfill$\diamondsuit$
\paragraph{Remark.}
Note that we did not assume $w\in C^0(\overline{\cal G})$ or
$C^0(\overline{\cal S})$. In fact, in our applications the function $w$ will be a
concavity function (see below), and the concavity function associated to an
unbounded function is not well defined on the boundary.
\section{Known convexity results for parabolic equations}
Let us recall that, in a classical paper \cite{BL}, Brascamp and Lieb
showed that the heat equation preserves log-concavity of the initial data.
Their method is based on analyzing the heat kernel representation and on
using the Brunn-Minkowsky inequality. In this paper we use different
methods, which are based on the maximum principle, and which are known to
yield an alternative proof of Brascamp and Lieb's result (see \cite{Koo}).
Korevaar (\cite{Ko}, \cite{Koo}) introduced the
elliptic and the parabolic {\it concavity function}, which may be defined
as
%
\begin{eqnarray}
&C(x,y)=v(z)-{\displaystyle v(x)+v(y)\over\displaystyle 2}&\\
&\mbox{or}&\nonumber\\
&C(x,y,t)=v(z,t)-{\displaystyle v(x,t)+v(y,t)\over\displaystyle 2},&
\label{parabolicCF}
\end{eqnarray}
%
respectively, where $z=(x+y)/2$. If $v(x)$ is a {\it continuous} function
on a convex domain $\Omega$, then $v$ is convex if and only if $C(x,y)\le
0$ in $\Omega^2$. Correspondingly if $v(x,t)$ is a {\it continuous}
function on a convex cylinder $\Omega\times(0,T)$, then it is convex w.r.t.\
$x$ for every fixed $t\in(0,T)$ if and only if $C(x,y,t)\le 0$ in
$\Omega^2\times(0,T)$.
By means of \eq{parabolicCF}, Korevaar investigates in \cite{Koo} the
equation
\begin{equation}\label{KE}
v_t=a^{ij}(t,\nabla v)\,v_{ij}-b(x,t,v,\nabla v)
\quad\mbox{in $\Omega\times(0,T)$},
\end{equation}
where $\nabla v$ is the spatial gradient. Under structural assumptions on
$(a^{ij})$ and $b$ he proves that $C$ can attain a positive maximum, if it
exists, only on the parabolic boundary of $\Omega^2\times(0,T)$, i.e. for
$t=0$ or when $x$ or $y\in\partial\Omega$. The structural assumptions were
$(a^{ij})>0$, $b_v\ge 0$, and concavity of $b$ with respect to $(x,v)$.
Of course, when dealing with a parabolic problem, one may also consider
convexity of the solution with respect to the $(N+1)$-dimensional variable
$(x,t)$. This leads to a study of the function
$$
C(x,y,t,s)=v(z,r)-{\displaystyle v(x,t)+v(y,s)\over\displaystyle 2},
$$
where $z$ is as before and $r=(t+s)/2$. { Porru and Serra} \cite{PS} use
this function to investigate equation \eq{KE} but with $(a^{ij})$
independent of $t$. They prove a convexity maximum principle of parabolic
type, i.e., the function $C(x,y,t,s)$ cannot attain a positive maximum
when $(x,t)$ and $(y,s)$ are not on the parabolic boundary of
$\Omega\times(0,T)$. This conclusion is reached under the following
structural assumptions on \eq{KE}:
\begin{enumerate}\itemsep=0pt
\item parabolicity, but not strict parabolicity of the equation;
\item boundedness of $b_v$ from below;
\item concavity of $b$ with respect to $(x,t,v)$.
\end{enumerate}
%
Kennington \cite{Kee} and Kawohl \cite{Ka}
also consider $C(x,y,t,s)$, but they regard the parabolic equation
as a degenerate elliptic equation in ${\mathbb R}^{N+1}$, and apply the
corresponding concavity maximum principle by Kennington \cite{Ke}.
They prove convexity of level sets of the solution to initial value
problems of the form
\begin{eqnarray*}
&u_t=\Delta u-f(u) \quad\mbox{in } \Omega\times{\mathbb R}^+,& \\
&u=1 \quad\mbox{on } \partial \Omega\times{\mathbb R}^+,& \\
&u(x,0)=1 \quad\mbox{in } \Omega ,&
\end{eqnarray*}
by applying the concavity maximum principle to a suitable transformed
fuction $v:=g(u)$. The result follows under suitable assumptions on $f$.
Due to the elliptic nature of the technique, an
estimate of $C$ is needed also for $t\to+\infty$.
\section{Space concavity}
\subsection*{A concavity maximum principle.}
In this subsection we generalize Korevaar's result cited above, concerning
convexity in space, in the sense that we drop the assumption of strict
parabolicity of the equation and we require $b$ to satisfy a Lipschitz
condition from below instead of $b_v\ge 0$. We do not require any other
smoothness of $(a^{ij})$ and $b$. After this extension, the result can
also be more easily compared to Porru
and Serra's result of convexity in space-time.
The function $b(x,t,v,p)$ is said to satisfy a {\it Lipschitz condition
from below} with respect to $v$ if there exists a constant $L\in{\mathbb R}$ such
that
%
\begin{equation}\label{LB}
b(x,t,u,p)-b(x,t,v,p)\ge -L\,(u-v)
\end{equation}
%
for all $x,t,u,v,p$ such that $u>v$.
\begin{theorem}\label{ConcavityMP}
Let $v\in C^2({\cal S})$ be a solution of\/ \eq{KE} in the infinite cylinder
${\cal S}:=\Omega\times{\mathbb R}^+$, $\Omega$ a convex bounded domain in ${\mathbb R}^N$. Suppose
$(a^{ij})\ge 0$ and let $b$ satisfy \eq{LB} and be concave in $(x,v)$.
\\
If\/ $\limsup C(x,y,t)\le 0$ as
$(x,y,t)\to\partial(\Omega^2\times{\mathbb R}^+)$,
then $C\le 0$ in all of\/ $\Omega^2\times{\mathbb R}^+$.
\end{theorem}
\paragraph{Proof.}
We construct a parabolic inequality for $C(x,y,t)$ in the domain
$\Omega^2\times{\mathbb R}^+$, and the result will follow from the classical maximum
principle (Corollary \en{MP}). A similar technique was used in \cite{GP}
for the elliptic case. Korevaar, instead, argued by contradiction at a
point where $C$ attained a positive maximum.
Let $\tilde A=(\tilde a^{hk}(x,y,t))$ be the $2N\times 2N$ matrix given by
$$
\tilde A=
\pmatrix{A & A\cr
A & A\cr},
$$
where $A=(a^{ij}(t,\nabla v(z,t)))$. By differentiating $C$ we find:
\begin{eqnarray*}
\tilde a^{hk}\,C_{hk}&=&a^{ij}(t,\nabla v(z,t))\,v_{ij}(z,t)
-a^{ij}(t,\nabla v(z,t))\,v_{ij}(x,t)/2\\
&&-a^{ij}(t,\nabla v(z,t))\,v_{ij}(y,t)/2,
\end{eqnarray*}
where $i,j=1,\dots,N$, $h,k=1,\dots,2N$ and the summation convention is in
effect. Since $\nabla v(z,t)-\nabla v(x,t)=2\,\nabla\!_x\,C(x,y,t)$, and
$\nabla v(z,t)-\nabla v(y,t)=2\,\nabla\!_y\,C(x,y,t)$, we may write
\begin{eqnarray}
\tilde a^{hk}\,C_{hk}+b^h\,C_h&=&a^{ij}(t,\nabla v(z,t))\,v_{ij}(z,t)
-a^{ij}(t,\nabla v(x,t))\,v_{ij}(x,t)/2 \nonumber \\
&&-a^{ij}(t,\nabla v(y,t))\,v_{ij}(y,t)/2, \label{algebraic}
\end{eqnarray}
where the (not necessarily bounded) coefficients $b^h$ for $h=1,\dots,N$
are given by
\begin{eqnarray*}
b^h(x,y,t)&=&{\displaystyle v_h(z,t)-v_h(x,t)
\over\displaystyle|\nabla v(z,t)-\nabla v(x,t)|^2} \times \\
&&\sum_{i,j=1}^N\
\Big(a^{ij}(t,\nabla v(z,t))-a^{ij}(t,\nabla v(x,t))\Big)v_{ij}(x,t)
\end{eqnarray*}
if $\nabla v(z,t)\ne\nabla v(x,t)$, and $b^h=0$ if $\nabla v(z,t)=\nabla
v(x,t)$. The expression of $b^h$ when $h=N+1,\dots,2\,N$ is analogous.
It is worthwile to stress the fact that equality holds in \eq{algebraic}
for algebraic reasons, and smoothness of $a^{ij}$ is by no means involved.
Now we may use equation \eq{KE} and obtain
\begin{eqnarray*}
\tilde a^{hk}\,C_{hk}+b^h\,C_h&=&v_t(z,t)+b(z,t,v(z,t),\nabla v(z,t))\\
&& - v_t(x,t)/2-b(x,t,v(x,t),\nabla v(x,t))/2\\
&& - v_t(y,t)/2-b(y,t,v(y,t),\nabla v(y,t))/2.
\end{eqnarray*}
%
On this expression we operate as follows:
\begin{enumerate}\itemsep=0pt
\item [i)] We replace $v_t(z,t)-v_t(x,t)/2-v_t(y,t)/2$ by $C_t(x,y,t)$.
\item [ii)] We replace $\nabla v(x,t)$ and $\nabla v(y,t)$, which appear as
arguments of $b$, by $\nabla v(z,t)$. This needs some
modification of the coefficients $b^h$, which is not relevant
for the conclusion (the technique is the same as before).
\item [iii)] We replace $b(z,t,v(z,t),\nabla v(z,t))$ by
\begin{eqnarray*}
b(z,t,(v(x,t)+v(y,t))/2,\nabla v(z,t))+c\,C(x,y,t),
\end{eqnarray*}
where $c$ is a suitable function, which is bounded from below
by virtue of \eq{LB}.
\end{enumerate}
%
Taking into account the concavity of $b$ with respect to $(x,v)$,
we obtain the following parabolic inequality:
$$
\tilde a^{hk}\,C_{hk}+b^h\,C_h\ge C_t+c\,C.
$$
Therefore the claim follows from Corollary \en{MP}.
\hfill$\diamondsuit$
\subsection*{Log-concavity in space.}\label{log-concavity}
We apply the concavity maximum principle proved above and study spatial
log-concavity of positive solutions to the following problem:
\begin{eqnarray}
& u_t=a^{ij}(t)\,u_{ij}-f(t,u,\nabla u)
\quad\mbox{in }{\cal S}:=\Omega\times{\mathbb R}^+, &\nonumber \\
& u(x,0)=u_0(x), & \label{problem} \\
& u(x,t)=0 \quad\mbox{for } x\in\partial\Omega,& \nonumber
\end{eqnarray}
where the coefficients $a^{ij}$ are real valued functions on ${\mathbb R}^+$
satisfying $(a^{ij})\ge 0$, $u_0\in C^0(\overline\Omega)$ is a given
log-concave function vanishing on $\partial\Omega$, and $\Omega$ is a
strictly convex bounded domain in ${\mathbb R}^N$, in the sense that
$\partial\Omega$ is of class $C^2$ and has positive Gauss curvature.
As usual, we say that $u_0$ is {\it log-concave} if it is {\it positive} in
$\Omega$ and $-\log u_0(x)$ is convex. We set $v(x,t):=-\log
u(x,t)$ and derive from \eq{problem} an equation satisfied by $v$. Then we
show that $v(x,t)$ is convex in $x$ for every given $t$, provided $f$
satisfies suitable conditions.
In order to obtain such a conclusion by means of the concavity maximum
principle we need to know that $v$ is convex near $\partial\Omega$. This
follows from the boundary conditions imposed on $u$, through the noteworthy
properties of the $\log$ function, and thanks to the strict convexity of
the domain $\Omega$, provided that
\begin{equation}
\label{gradient}
\nabla u(x,t)\ne 0 \quad \mbox{for all $x\in\partial\Omega$ and $t\ge 0$.}
\end{equation}
More precisely, a function $u(x)$ which is positive in a strictly convex
domain $\Omega$, and which vanishes on $\partial\Omega$ with nonvanishing
gradient there, is always concave (hence log-concave) near $\partial\Omega$
in the tangential directions. On the other side, since $\nabla u\ne 0$ on
$\partial\Omega$, $u$ is also log-concave near $\partial\Omega$ in the
normal direction. The interested reader may consult \cite{Koo}, Lemma 2.4,
or \cite{GP}, Lemma 3.2, for details.
The final conclusion is the following:
\begin{theorem} \label{general}
Let $u\in C^2({\cal S})\cap C^1(\overline{\cal S})$ be a positive classical solution
of problem\/ \eq{problem}, where $u_0$ and\/ $\Omega$ are as above.
Assume\/ \eq{gradient}. If $f(t,u,p)$ is of class
$C^2({\mathbb R}^+\times{\mathbb R}^+\times{\mathbb R}^N)$ and such that
\begin{eqnarray}
f_u+\frac{p\cdot\nabla\!_p\,f}u-\frac fu & \ge & -L, \label{first} \\[2pt]
f_u+\frac{p\cdot\nabla\!_p\,f}u-\frac fu & \le &
u\,f_{uu}+2\,p\cdot\nabla\!_p\,f_u+\frac{p_i\,p_j\,f_{p_ip_j}}u,
\label{second}
\end{eqnarray}
for every $t>0$, $u>0$, $p\in{\mathbb R}^N$, and with a suitable constant $L>0$,
then $u$ is log-concave in space for all $t$.
\end{theorem}
\paragraph{Proof.}
By computation we find that $v(x,t):=-\log u(x,t)$ satisfies the equation
$v_t=a^{ij}(t)\,v_{ij}-b(t,v,\nabla v)$, where
$$
b(t,v,\nabla v)=a^{ij}(t)\,v_i\,v_j-e^v\,f(t,e^{-v},-e^{-v}\,\nabla v).
$$
Furthermore, the derivative $b_v$ is given by
$$
b_v(t,v,\nabla v)=f_u(t,e^{-v},p)
+e^v\,p\cdot\nabla\!_p\,f(t,e^{-v},p)-e^v\,f(t,e^{-v},p),
$$
where we set, for shortness, $p:=-e^{-v}\,\nabla v$, therefore it is
bounded from below provided \eq{first} holds. Finally,
we have
$$
b_{vv}(t,v,\nabla v)=f_u+\frac{p\cdot\nabla\!_p\,f}u-\frac fu
-u\,f_{uu}-2\,p\cdot\nabla\!_p\,f_u
-\frac{p_i\,p_j\,f_{p_ip_j}}u,
$$
where $f$ and its derivatives are all evaluated at $(t,e^{-v},p)$. Hence
by \eq{second} we have $b_{vv}\le 0$ and the assumptions of Theorem
\en{ConcavityMP} are satisfied. As remarked above, the parabolic concavity
function associated to $v$ satisfies $\limsup C(x,y,t)\le 0$ as
$(x,y,t)\to\partial(\Omega^2\times{\mathbb R}^+)$. This and the concavity maximum
principle imply $C(x,y,t)\le 0$ for all $(x,y,t)\in\Omega^2\times{\mathbb R}^+$.
\hfill$\diamondsuit$
\paragraph{Remarks.}
\begin{enumerate}
\item We have preferred to state the theorem in an infinite cylinder for
the sake of simplicity. Of course the result holds, {\it mutatis mutandis},
as long as the solution exists.
\item Let us just mention that a solution $u$ of \eq{problem} has to be
{\it positive} in $\Omega\times{\mathbb R}^+$, and it has to satisfy \eq{gradient},
provided $f_u$ is bounded from below and one of these conditions is
verified:
\begin{enumerate}\itemsep=0pt
\item The equation in \eq{problem} is uniformly parabolic, $f(t,0,0)\le
0$ for all $t>0$, $\nabla\!_p\,f$ is bounded, and $\nabla u_0\ne 0$
on $\partial\Omega$. In this case the null constant turns out to be a
subsolution of the same problem, therefore $u$ must be positive by the
strong comparison principle, and $\nabla u$ does not vanish on $\Omega$
by Hopf's Lemma.
\item The function $f$ does not depend on $t$, and there exists a
positive solution $w(x)$ of the stationary problem $\Delta w=f(w,\nabla w)$
satisfying $w\le u_0$, $\nabla w\ne 0$ on $\partial\Omega$. In this case
$w(x,t):=w(x)$ is a subsolution.
\end{enumerate}
\item \label{approx} Theorem \en{general} still holds with a log-concave
$u_0$ whose gradient vanishes at the boundary, provided we can
approximate $u_0$ in the $C^0$-norm by a sequence of log-concave $u_{0k}$
satisfying $\nabla u_{0k}\ne 0$ on $\partial\Omega$ and such that the
corresponding problems \eq{problem} have positive solutions $u_k$ in
${\cal S}$. This can be seen as follows. Since $u_k$ is log-concave,
$$
u_k^2\!\left(\frac{x+y}2,t\right)\ge u_k(x,t)\,u_k(y,t)
\quad\mbox{in $\Omega^2\times{\mathbb R}^+$.}
$$
This inequality is preserved under the pointwise limit as
$k\to+\infty$. Since $u$ is supposed to be positive, the corresponding
inequality for $u$ is equivalent to log-concavity.
\item For the same reason, and under similar conditions, we can even take
$u_0\in C^0(\Omega)$, i.e., not continuous up to the boundary.
\item The result can be extended to the case when $\Omega$ is convex but
not strictly convex by means of a similar argument, provided the solution
$u$ of problem \eq{problem} is positive. It suffices that there exists a
sequence of strictly convex domains $\Omega_k$ and log-concave initial
data $u_{0k}$ to which Theorem \en{general} is applicable, and such that
the corresponding solutions $u_k$ converge pointwise to $u$.
\item It is always possible to approximate a log-concave $u_0$ in the
$C^0$-norm by a sequence of log-concave $u_{0k}$ satisfying $\nabla
u_{0k}\ne 0$ on the boundary. See the appendix for an explanation.
\end{enumerate}
\subsection*{Examples and applications.}
Of course, the assumptions of Theorem \en{general} are satisfied
by the linear heat equation $u_t=\Delta u$. More generally, if $f$ does
not depend on the gradient, then \eq{first}-\eq{second} reduce to
\begin{equation}\label{reduced}
-L\le f_u-f/u\le u\,f_{uu},
\end{equation}
which is satisfied, for instance, by the equation $u_t=\Delta u-u^\alpha$
with $\alpha\ge 1$, and by $u_t=\Delta u-u\,\log^\beta(1+u)$ with $\beta\ge
1$.
We may ask whether, for some special $f$, the two
equalities hold in \eq{reduced}. The answer is positive: indeed, the
function $f(u):=(m-\log u)\,u+q$, $m,q\in{\mathbb R}$, satisfies
$-1=f_u-f/u=u\,f_{uu}$. Theorem \en{general} is therefore applicable to
the corresponding equations, and in particular to
$$
u_t=\Delta u+u\,\log u.
$$
Note that the cited result by Korevaar is not applicable to such
equation, because in this case $b_v$ is negative.
In case $f=g(u)+h(|\nabla u|)$ then \eq{first}-\eq{second} hold provided
$g(s)$ and $h(s)$ satisfy
\begin{eqnarray*}
&-L\le g'-g/s\le s\,g'',&\\
& 0\le h'-h/s\le s\,h''.&
\end{eqnarray*}
If, in particular, $f=u^\alpha+|\nabla u|^\beta$ then Theorem \en{general}
is not directly applicable for $\beta<2$, since $f$ does not meet the
smoothness requirements. However, an inspection of the proof shows that the
conclusion still holds provided $\alpha,\beta\ge 1$.
Similarly, one sees that for $f=u^\alpha\,|\nabla u|^\beta$ the conclusion
of Theorem \en{general} holds when $\alpha+\beta\ge 1$.
Let us remark that we admit the dependence on time and the degeneracy of
the matrix $(a^{ij})$, so that Theorem \en{general} is applicable to
equations like $u_t=(1+\sin t)\,\Delta u$.
Once we know that the solution to \eq{problem} is log-concave in space, we
deduce that {\it the level sets $E(c,t):=\{\,x\in\Omega\mid u(x,t)\ge
c\,\}$ are convex} for all $c$ and $t$. In particular, the set where
$u(\cdot,t)$ attains its maximum (the {\it hot spot}) is convex for every
given $t$. If $u$ blows up at a finite time $T$, then the set where
$u(x,T)=+\infty$ is convex.
\section{Further remarks and open problems}
Let us remark that there are some interesting equations to which our result
does not apply. This is due to different factors. For instance, the
following equations do not satisfy \eq{first}-\eq{second}:
\begin{eqnarray*}
&&u_t=\Delta u+u^\alpha \quad\mbox{(with the $+$ sign);}\\
&&u_t=\Delta u+e^u;\\
&&u_t=\Delta u+u^\alpha\,\log u \quad\mbox{with $\alpha\ne 1$ and positive;}\\
&&u_t=\Delta u+u^\alpha-|\nabla u|^\beta, \quad\alpha,\beta>0;\\
&&u_t=\Delta u-u^\alpha+|\nabla u|^\beta, \quad\alpha,\beta>0.
\end{eqnarray*}
The equation $u_t=\Delta u-e^u$, instead, satisfies \eq{first}-\eq{second}
but the null constant is a supersolution (not a subsolution), and the
stationary solution with homogeneous Dirichlet boundary conditions is
negative.
We also want to point out that the transformation $v:=-\log u$ is not the
only conceivable one to investigate convexity of the level sets of
a positive $u$, but it turns out to be particularly well-featured for that
purpose. Among the other transformations that are used, one of the most
simple is given by $v=-u^\alpha$ with $\alpha\in(0,1)$. This leads to the
notion of {\it power concavity} (see \cite{Ke}, \cite{Kaa},
\cite{Kaaa}). Even the general transformation $v:=g(u)$, with a decreasing
and convex $g$, has been investigated (\cite{Koo}, \cite{GP}, \cite{Kaa}).
Therefore one may ask what happens if we consider problem \eq{problem} with
an initial datum which is, for instance, power concave. We have tried to
derive an equation for $v:=-u^\alpha$ with $\alpha\in(0,1)$ and to apply
the concavity maximum principle (Theorem \en{ConcavityMP}). Unfortunately,
even for the special case $u_t=\Delta u-f(u)$, it turns out that the
equation for $v$, which takes the form $v_t=\Delta v-b(v,\nabla v)$ with
$$
b(v,\nabla v)=\frac{\alpha-1}{\alpha\,v}\,|\nabla v|^2
-\alpha\,(-v)^{1-1/\alpha}\,f((-v)^{1/\alpha}),
$$
does not satisfy the assumptions of Theorem \en{ConcavityMP}, since the
first term in $b$ is convex with respect to $v$ for $v<0$. Similarly, the
substitution $v=u^\alpha$ with $\alpha<0$ leads to
$$
v_t=\Delta v-\frac{\alpha-1}{\alpha\,v}\,|\nabla v|^2
-\alpha\,v^{1-1/\alpha}\,f(v^{1/\alpha}),
$$
and again the assumptions of Theorem 4.1 are violated.
\section{Appendix: an approximation lemma}
The purpose of assumption \eq{gradient} in Theorem \en{general} is that of
ensuring spatial log-concavity of the solution $u$ near the parabolic
boundary. If we only require that $\nabla u(x,t)\ne 0$ for
$x\in\partial\Omega$ and $t>0$, i.e., if we admit $\nabla u(z_0,0)=0$ for
some $z_0\in\partial\Omega$, then we cannot {\it a priori\/} exclude the
existence of a sequence $(x_k,y_k,t_k)$ such that $x_k,y_k\to
z_0\in\partial\Omega$, $t_k\to 0$ and $C(x_k,y_k,t_k)>\varepsilon>0$ for
all $k$.
Such kind of {\it corner pathology\/} is quite subtle and it is sometimes
neglected. For instance, Lemma 3.11 in \cite{Kaa} did not take into
consideration an analogous question for the elliptic case, and was later
fixed in \cite{GP}, Lemma 3.2. The paper by Korevaar \cite{Koo} also does
not discuss this point in the parabolic case.
As remarked before (Section \en{log-concavity}, Remark \ref{approx}), a
possibility to overcome this difficulty consists of approximating the
initial datum $u_0$, which is supposed to be log-concave but whose gradient
may now vanish on the boundary, by a sequence of log-concave $u_{0k}$ that
satisfy $\nabla u_{0k}\ne 0$ on $\partial\Omega$.
In this appendix we show that such an approximation is always possible.
\begin{lemma}
Let\/ $\Omega$ be a convex (but not necessarily strictly convex) bounded
domain in\/ ${\mathbb R}^N$. Let $u_0\in C^0(\overline\Omega)$ be a log-concave
function in\/ $\Omega$ that vanishes on $\partial\Omega$. In order to
simplify the presentation, we assume that $\partial\Omega$ is of class
$C^1$ and that $u_0\in C^1(\Omega)$.
\\
Then there exists a sequence of log-concave $u_{0k}$ such that
\begin{enumerate}
\item [\rm(a)] $\sup_\Omega|u_{0k}(x)-u_0(x)|\to 0$ as $k\to+\infty$;
\item [\rm(b)] $u_{0k}(x)\equiv a_k\,\mbox{\rm dist$(x,\partial\Omega)$}$
for some $a_k>0$ and all $x$ sufficiently close to $\partial\Omega$;
\item [\rm(c)] in particular, $\nabla u_{0k}\ne 0$ on $\partial\Omega$ for
each $k$.
\end{enumerate}
\end{lemma}
\paragraph{Proof.}
The structure of the proof is the following. We consider the function
$v(x):=-\log u_0(x)$, which is convex and unbounded from above by
assumption. For each integer $k>\min_\Omega v$, we modify $v$ near
$\partial\Omega$ and we obtain a suitable $v_k$. Finally, we show that
$u_{0k}:=e^{-v_k(x)}$ verifies (a)-(b). Of course, (c) is a consequence of
(b).
Define $\Omega_k:=\{\, x\in\Omega \mid v(x)>k \,\}$. For $k$ large,
$\Omega_k$ is nonempty and has a smooth boundary $\partial\Omega_k$ whose
distance from $\partial\Omega$ is positive. For each
$y\in\partial\Omega_k$ we consider the tangent plane to the graph of $v$ at
$y$. Such tangent plane is the graph of a function that we denote by
$\pi_y(x)$. Define $\tilde v_k$ on $\Omega$ by setting
$$
\tilde v_k(x)=\cases{
v(x), &if $x\in\Omega_k$;
\cr\noalign{\medskip}
\sup\limits_{y\in\partial\Omega_k}\pi_y(x),
& if $x\in\Omega\setminus\Omega_k$.
\cr}
$$
Then $\tilde v_k$ is a convex function attaining a {\it finite} value on
$\partial\Omega$. Now choose $b_k\in{\mathbb R}$ so large that $\tilde
v_k(x)=v(x)\ge -\log\mbox{dist$(x,\partial\Omega)$})-b_k$ in $\Omega_k$. For instance, we may
take $b_k:=-\min_\Omega v-\log\mbox{dist$(\Omega_k,\partial\Omega)$}$.
Define
$$
v_k(x):=\max\Big(\tilde v_k(x),-\log\mbox{dist$(x,\partial\Omega)$}-b_k\Big),
$$
which is convex because the function $\mbox{dist$(x,\partial\Omega)$}$ is
concave, hence log-con\-cave, and the maximum of two convex functions is
still convex.
Since $\tilde v_k$ is bounded on all of $\Omega$, we have
$v_k(x)=-\log\mbox{dist$(x,\partial\Omega)$}-b_k$ near $\partial\Omega$ and
therefore the function $u_k(x):=e^{-v_k(x)}$ coincides with
$a_k\,\mbox{dist$(x,\partial\Omega)$}$ for
$\mbox{dist$(x,\partial\Omega)$}$ small and $a_k:=e^{b_k}$.
It remains to check (a). By definition, $u_{0k}$ coincides with $u_0$ in
$\Omega_k$. Outside $\Omega_k$ we have $v_k(x)\ge\tilde v_k(x)\ge k$, hence
$u_k(x)\le e^{-k}$. This implies $\sup_\Omega|u_{0k}(x)-u_0(x)|\le
e^{-k}+\sup_{\Omega\setminus\Omega_k}|u_0(x)|$. Since $\Omega_k$ invades
$\Omega$ as $k\to+\infty$, and since $u_0$ is supposed to vanish on
$\partial\Omega$, claim (a) follows and the proof is complete.
\hfill$\diamondsuit$
\begin{thebibliography}{99}
\bibitem{ALL}
{\sc O. Alvarez, J.-M. Lasry, P.-L. Lions},
{\it Convex viscosity solutions and state constraints}, J. Math. Pures Appl.
{\bf 76} (1997), 265--288.
\bibitem{BL}
{\sc H. J. Brascamp, E. H. Lieb},
{\it On extensions of the Brunn-Minkowsky and Pr\'ekopa-Leindler theorems,
including inequalities for log concave functions, and with an
application to the diffusion equation}, J. Funct. Anal.
{\bf 22} (1976), 366--389.
\bibitem{CF}
{\sc L. A. Caffarelli, A. Friedman},
{\it Convexity of solutions of semilinear elliptic equations}, Duke Math. J.
{\bf 52} (1985), 431--456.
\bibitem{GP}
{\sc A. Greco, G. Porru},
{\it Convexity of solutions to some elliptic partial differential equations},
SIAM J. Math. Anal.
{\bf 24} (1993), 833--839.
\bibitem{Ka}
{\sc B. Kawohl},
{\it Convexity in the parabolic dead core problem}, unpublished manuscript.
\bibitem{Kaa}
{\sc B. Kawohl}, Rearrangements and convexity of level sets in PDE,
Lecture Notes in Math. {\bf 1150}, Springer-Verlag 1985.
\bibitem{Kaaa}
{\sc B. Kawohl},
{\it When are solutions to nonlinear elliptic boundary value problems
convex?} Comm. Partial Differential Equations
{\bf 10} (1985), 1213--1225.
\bibitem{Ke}
{\sc A. U. Kennington},
{\it Power concavity and boundary value problems}, Indiana Univ. Math. J.
{\bf 34} (1985), 687--704.
\bibitem{Kee}
{\sc A. U. Kennington},
{\it Convexity of level curves for an initial value problem},
J. Math. Anal. Appl.
{\bf 133} (1988), 324--330.
\bibitem{Ko}
{\sc N. J. Korevaar},
{\it Capillary surface convexity above convex domains}, Indiana Univ. Math. J.
{\bf 32} (1983), 73--82.
\bibitem{Koo}
{\sc N. J. Korevaar},
{\it Convex solutions to nonlinear elliptic and parabolic boundary value
problems}, Indiana Univ. Math. J.
{\bf 32} (1983), 603--614.
\bibitem{PS}
{\sc G. Porru, S. Serra},
{\it Maximum principles for parabolic equations},
J. Austral. Math. Soc. Ser. A
{\bf 56} (1994), 41--52.
\bibitem{PW}
{\sc M. H. Protter, H. F. Weinberger},
Maximum principles in differential equations,
Prentice Hall, Englewood Cliffs, N.J., 1967.
\bibitem{RR}
{\sc M. Renardy, R. C. Rogers}, An introduction to partial differential equations,
Springer-Verlag 1993.
\end{thebibliography}
\medskip
\noindent{\sc Antonio Greco} \\
Dipartimento di Matematica, Universit\`a di Cagliari \\
via Ospedale 72 \\
I 09124 Cagliari, Italy \\
E-mail: greco@vaxca1.unica.it \\
http://riemann.unica.it/$\sim$antoniog
\medskip
\noindent{\sc Bernd Kawohl} \\
Mathematisches Institut, Universit\"at zu K\"oln \\
D 50923 K\"oln, Germany \\
E-mail: kawohl@mi.uni-koeln.de \\
http://www.mi.uni-koeln.de/mi/Forschung/Kawohl
\end{document}