\documentstyle[twoside]{article}
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\markboth{\hfil ONE-SIDED MULLINS-SEKERKA FLOW\hfil EJDE--1993/08}% 
{EJDE--1993/08\hfil U. F. Mayer\hfil}
\begin{document}
\ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations}\newline
Vol. {\bf 1993}(1993), No. 08, pp. 1-7. Published December 13, 1993.\newline
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113 }
\vspace{\bigskipamount}\\ 
One-sided Mullins-Sekerka Flow Does Not Preserve Convexity
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35R35, 
35J05, 35B50, 53A07.\newline\indent
{\em Key words and phrases:} Mullins-Sekerka flow, Hele-Shaw flow,
Cahn-Hilliard equation,\newline\indent
 free boundary problem, convexity, curvature.\newline\indent
\copyright 1993 Southwest Texas State University  and University of 
North Texas\newline\indent
Submitted: November 6, 1993.} }
\date{}
\author{Uwe F. Mayer} 
\maketitle

\begin{abstract} 
The Mullins-Sekerka model is a nonlocal evolution model for  
hypersurfaces,
which arises as a singular limit for the Cahn-Hilliard equation.
Assuming the existence of sufficiently smooth solutions we will show
that the one-sided Mullins-Sekerka flow does not preserve  
convexity.  \end{abstract}
\def\real{{\bf R}}
\def\blank{\mbox{}\\}
\newtheorem{thm}{Theorem}
\newtheorem{prop}{Proposition}

\section*{Introduction}
The Mullins-Sekerka flow is a nonlocal generalization of the mean
curvature flow arising from physics \cite{MS,P}. Similar to
Stefan-type problems there is a one-sided
and a two-sided version. Recently it has been shown rigorously that
the two-sided model arises as a
singular limit of the Cahn-Hilliard equation \cite{ABC}. This has  
been known
formally since the work of Pego \cite{P}. In the literature the
Mullins-Sekerka model has been often called Hele-Shaw model. However,
there are two different problems which are called Hele-Shaw problems,
compare for example \cite{ABC,C} with \cite{CP}. The problem studied  
in this paper is the same as the one-sided version of the Hele-Shaw 
problem as formulated in \cite{ABC,C}.
To avoid this confusion one should probably call the Hele-Shaw flow  
of \cite{ABC,C} the Mullins-Sekerka flow.

One can ask whether the properties of the mean
curvature flow can be generalized to the Mullins-Sekerka flow.
Not all results can be expected to generalize, due to the
nonlocal character of the Mullins-Sekerka problem, in particular not  
those
that rest on a local argument for the mean curvature flow.  There has
been some progress made towards the question of existence, see  
\cite{DR} for
the one-sided version and \cite{C} for the two-sided version.
Recently Luckhaus has announced further results concerning existence,
however, no details are know by the author. It is known that the mean
curvature flow preserves convexity \cite{GH,H}. It is therefore a
natural question 
to ask whether this is also true for the Mullins-Sekerka flow. Under  
the assumption of short-term existence of sufficiently smooth solutions
this question  
is answered negatively in this paper for the one-sided case.

\section*{The 2-dimensional case}
We look at a curve $\Gamma_0$\/ and the free boundary problem  
governed by the evolution law given by
\begin{equation}
\left\{ \begin{array}{rlll}
       \Delta u & = & 0 & \mbox{inside of }\Gamma_t\,, \\
       u & = & \kappa & \mbox{on }\Gamma_t\,, \\
       v & = & -\frac{\partial u}{\partial n}\,.
       \end{array} \right. \end{equation}
Here $n$\/ is the {\em outer}\/ unit normal to $\Gamma_t$, $v$\/ and
$\kappa$\/ are the normal velocity and the curvature of $\Gamma_t$,
respectively. The signs are chosen in such a way that a circle has
positive curvature, and a shrinking curve has negative velocity.

The principal idea is to look at a shape given by a straight tube  
with two circular end caps.
By the strong maximum principle $\frac{\partial u}{\partial n}>0$\/  
on the circular parts of $\Gamma_0$. Let us restrict our attention to  
the right part of the figure. $\frac{\partial u}{\partial n}>0$\/ implies
$u_x>0$\/ on the circular path. On the straight part we have
$u_x\equiv 0$, as $u$\/ is identically zero there. We also have
$u_x\equiv 0$\/ on the $y$-axis by symmetry for $u$. Ignoring for the
moment the discontinuity of $u_x$ we conclude $u_x>0$\/ in the
interior of the right half by the maximum principle. 


As $u_x\equiv 0$\/ on the (upper) straight line, we must have
$\frac{\partial u_x}{\partial n} =u_{xy}<0$\/ on the right half of it
by another application of the maximum principle. Even another
application of the maximum principle for the function $u$\/ tells us
that $\frac{\partial u}{\partial n}=u_y<0$\/ on the upper straight
line. Therefore on the right half of the upper line $|u_y|$\/
decreases towards the center. By symmetry we get the distribution of
the initial speed sketched in Figure 1.


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\caption{Distribution of initial velocity}
\end{figure}
Therefore the center will move out slower than the rest of the
straight line, the figure will evolve into a nonconvex shape. 


This example has one fatal flaw. While the straight lines want to  
move
out, the circular parts want to move in. This of course will break up
the curve instantaneously, and (1) cannot possibly be satisfied with
an initial configuration like this. The solution to the dilemma is
obviously to smoothen out the corners.
 

\medskip

{\em For the sequel we will assume that the one-sided Mullins-Sekerka
flow allows
a smooth solution provided the initial configuration $\Gamma_0$ is
$C^\infty$.}

\medskip

The difficulty for a smooth domain lies now in showing that $u_x$\/
still has the sign we want. On the parts of $\Gamma_0$\/ where
$u\equiv 0$\/ or $u\equiv 1$\/ we get the correct sign by the maximum
principle as before. However, the maximum principle does not help on
the transition parts. 

The following discussion is restricted to the right lower quarter of
$\Gamma_0$. Let $\gamma$\/ be the transition path from the straight
line to the circular part, and $\kappa :[0,L]\mapsto[0,1]$\/ be the
curvature on $\gamma$, parametrized by arc length. We choose
$\kappa$\/ to be a monotonous function. 

Then $\gamma$\/ is given by
\[\left\{ \begin{array}{rll}
x(s)&=& \int_0^s\cos \left( \int_0^\sigma \kappa (t)\, dt\right)\,
d\sigma+x_0\,, \\ 
&& \\
y(s)&=& \int_0^s\sin \left( \int_0^\sigma \kappa (t)\, dt\right)\,
d\sigma+y_0\,. 

\end{array}\right.\]
Let $\overline{\gamma}$\/ be the curve in 3-space over $\gamma$\/
parametrized by $(x(s),y(s),\kappa(s))$, and let
$\beta$\/ be the projection of  $\overline{\gamma}$\/ onto the  
$y$-$u$-plane.
\begin{prop} For a suitable choice of $\kappa$
the curve $\beta$ is concave down.
\end{prop} 

One can choose, for example, 
\[\kappa(s)=\frac{1}{C}\int_0^{s/L}e^\frac{1}{t(t-1)}\, dt\,,\  
\ s\in [0,L]\,,\] 
where $C$\/ is chosen to have $\kappa(L)=1$. The curve $\beta$\/ is
described by 
\[y(u)=\int_0^{\kappa^{-1}(u)}\sin \left( \int_0^\sigma \kappa (t)\,
dt\right)\, d\sigma+y_0\,. \] 
Concavity can be checked with methods from elementary calculus, the
rather technical details will be omitted here.

We show now how the proposition can be used to  construct upper
barriers for $u$. Pick any point $Q$\/ on $\overline{\gamma}$, and  
let
$B$\/ be the projection of $Q$\/ onto the $y$-$u$-plane. Let
$u=my+c$\/ be the equation of the tangent line of $\beta$\/ at $B$,
where $m$\/ and $c$\/ depend on $Q$. This equation defines a plane in
3-space. If $\overline{\Gamma_0}$\/ denotes the curve in 3-space over
the curve $\Gamma_0$\/ given by $(x,y,\kappa)$, then this plane
touches $\overline{\Gamma_0}$\/ in exactly two points, namely in  
$Q$\/
and in the symmetric image of $Q$\/ in left half of $\Gamma_0$.


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\end{picture}
\end{center}
\caption{The graphs of $\overline{\Gamma_0}$, $\beta$, and of
$u=my+c$}
\end{figure}

A nonvertical plane of course is the graph of an harmonic function.  
By
construction this plane has zero $x$-derivative everywhere. From
comparison we see that $u_x>0$\/ at $Q$. From here on we conclude the
argument as before. Therefore we have proved the following 

\begin{thm} Assume that {\rm(1)} allows a smooth solution
provided the initial configuration is $C^\infty$. Then there
are convex smooth initial configurations consisting of a straight  
tube with two
end caps that will evolve into nonconvex curves. The flat tube can be
arbitrarily short. In particular these initial curves can be chosen  
to be
arbitrarily close in the $C^1$-norm to a circle.
\end{thm} 

\section*{The $k$-dimensional case, $k\geq 3$}
We look at a hypersurface $\Gamma_0$\/ and the free boundary problem
governed by the evolution law given by 

\begin{equation}
\left\{ \begin{array}{rlll}
       \Delta u & = & 0 & \mbox{inside of }\Gamma_t\,, \\
       u & = & H & \mbox{on }\Gamma_t\,, \\
       v & = & -\frac{\partial u}{\partial n}\,.
       \end{array} \right. \end{equation}
Here $n$\/ is the {\em outer}\/ unit normal to $\Gamma_t$, $v$\/ and
$H$\/ are the normal velocity and the mean curvature of $\Gamma_t$,
respectively. The signs are chosen in such a way that a sphere has
positive curvature, and a shrinking surface has negative velocity.


We use the curve from the 2-dimensional case and rotate about the
$x$-axis. The resulting hypersurface in $\real^k$ is given by 
\[\Gamma_0=\left\{ (x(s),y(s)\omega)\,:\,s\in[0,L],\,\omega\in  
S^{k-2}
\subset \real^{k-1}\right\}\,.\] 

\begin{prop} The principal curvatures of $\Gamma_0$ are given by
\[\kappa_{1}=\kappa\,,\ \ \kappa_i=-\frac{x^\prime}{y}\,,\ \
i=2,\ldots,k-1\,.\]
\end{prop}
The proof involves only routine computations of the first and second
fundamental forms in local coordinates and will be omitted here.

The mean curvature is 
\[ H=\sum_{i=1}^{k-1}\kappa_i=\kappa-(k-2)\frac{x'}{ y}.\]
Let $\overline{\Gamma_0}=\{(x,u)\in\real^{k+1}\,:\,x\in  
\Gamma_0,\,u=H(x)\}$.
As before we project along the $x_1$-axis onto the
hyperplane in $\real^{k+1}$ perpendicular to the $x_1$-axis. The
resulting manifold $\beta$\/ is the graph of $u=H(x_2,\ldots, x_k)$. 

\begin{prop}
For a suitable choice of $\kappa$ the graph $\beta$ is concave down.
\end{prop}
The first step for the proof is to note that this graph is
rotationally symmetric with respect to $(x_2,\ldots, x_k)$\/ by
construction. Hence it is enough to look at a radial section, say, in
the $x_2$-$u$-plane. In the sequel we will use $y$\/ instead of  
$x_2$.
We have already seen that $u=\kappa(y)$\/ is concave down. The same  
is
true for $u=\kappa_i(y),\ i=2,\ldots,k-1$. These curves are given by
\[\left\{ \begin{array}{rll}
y(s)&=&\int_0^s\sin \left( \int_0^\sigma \kappa  
(t)\,dt\right)\,d\sigma+y_0\\ & & \\
\kappa_i(s)&=&\displaystyle\frac{\cos \left( \int_0^s \kappa
(t)\,dt\right)}{y(s)} 
\end{array}\right.\]
where the curvature $\kappa$ is chosen to be the same as in the
2-dimensional case. As $y(s)$ is increasing one can express  
$\kappa_i$
as a function of $y$\/ and then use methods from calculus to show
concavity.
Therefore $H$\/ is concave down. 

From here on we proceed exactly as in the 2-dimensional case. For a
given point $Q\in\overline{\Gamma_0}$ we look at the tangent
hyperplane to its projection $B\in\beta$\/ in
$x_2$-$\ldots$-$x_k$-$u$-space, and use the equation
$\overline{u}=a_2x_2+\ldots +a_kx_k+c$\/ of this hyperplane as the
definition of an affine function on $\real^k$. This gives us a
supersolution $\overline{u}$\/ to the harmonic function $u$\/ of (2)
connected with $\Gamma_0$. We get the same conclusion on the sign of
$u_{x_1}$\/ in the same way as in the 2-dimensional case. To proceed
we need the additional information that $u$\/ be symmetric about the
$x_1$-axis. This is true because of the symmetry of the domain, the
symmetry of the boundary data, and the invariance of the Laplacian
under rotations. Therefore it is enough to look at a section, say, in
the $x_1$-$x_2$-plane. There we have already seen how the information
on the sign of $u_{x_1}$\/ implies that $|u_{x_2}|=|\frac{\partial
u}{\partial n}|$\/ decreases towards the middle.
\begin{thm} Assume that {\rm(2)} allows a smooth solution
provided the initial configuration is $C^\infty$.
Then there are convex smooth initial configurations in $\real^k$  
consisting of
a cylinder with two end caps that will evolve into nonconvex
hypersurfaces. The cylinder can be arbitrarily short. In particular
these initial hypersurfaces can be chosen to be arbitrarily close in  
the $C^1$-norm to a sphere.
\end{thm} 

\noindent {\bf Remark.} The proof of the proposition shows that in
fact more is true. The only properties of $H$\/ we used are that the
arithmetic mean preserves concavity, and, for positive arguments, is
positive and increases in each argument. One can therefore replace
$H$\/ by any suitably smooth function that has these properties and
get the same results. In particular one can use $\sqrt[r]{H_r},$  
where
$H_r$\/ is the $r$-th symmetric function of the principal curvatures  
\cite{G}. 

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{\sc  Department of Mathematics, 
University of Utah,
Salt Lake City, Utah 84112}\par 
E-mail address: mayer\@@csc-sun.math.utah.edu

\end{document}