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\title{ Results and Questions on \\ Minimum Problems for Eigenvalues }
\author {\sc Dorin Bucur\thanks{CNRS-Equipe de Math\'ematiques, Universit\'e de
 Franche-Comt\'e, 16 route de Gray, 25030 Besan\c con Cedex, France, 
bucur@math.univ-fcomte.fr} \and \sc Giuseppe Buttazzo\thanks{Dipartimento di 
Matematica, Universit\`a di Pisa, Via Buonarroti 2,
56127 Pisa, Italy, buttazzo@dm.unipi.it} }
\date{}









 \begin{document}
\maketitle
\begin{abstract}
For every open subset $\Om$ of $\RR^N$ we consider the sequence 
$\lb(\Om)=(\lb_j(\Om))_j$ of eigenvalues of the Laplace operator 
$-\Delta$ on $H^1_0(\Om)$ and we consider the minimization problem
\[\min\{\Phi (\lb(\Om)) \; : \; |\Om|=m , \Om \in {\cal A}\}\]
where the function $\Phi$ is given and ${\cal A}$ denotes the class of
 admissible domains. About this problem, we present some known results 
and we list some open questions.
\end{abstract}

\section{Introduction}



Minimization problems for eigenvalues are not new in the literature; since
 the first result by Krahn \cite{kr24}, \cite{kr26} and Faber \cite{faber} 
concerning the minimality of a
disk in $\RR^2$ for the first eigenvalue of the Laplace operator $-\Delta$
 with Dirichlet boundary conditions, among domains with equal area, many other
 results have been obtained. 

We are concerned with problems of the form
\begin{equation}\label{ar11}
\min\{\Phi (\lb(\Om)) \; : \; |\Om|=m , \Om \in {\cal A}\}
\end{equation}
where $\lb(\Om)$ denotes the sequence $(\lb_j(\Om))_j$ of all eigenvalues of
 the Laplace operator with Dirichlet boundary conditions, $\Phi$ is a given 
function, $m>0$ is given and ${\cal A}$ is the class of admissible domains. A
 choice we can do is to take ${\cal A}=\{ \Om \sq D\}$ where the domain $D$ 
(the so called ``{\it design region}'') is given, a bounded open subset of 
$\RR^N$,
 or also 
${\cal A}=\{ \Om \sq \RR^N\}$ without any a priory imposed boundedness. We 
shall see that the two cases are quite different, and most of the results 
obtained for the first cannot be easily extended to the second.


The plan of the paper is the following.
\begin{itemize}
\item{ In section \ref{ar2} we present a general result concerning shape
 optimization problems with monotone cost functionals; we introduce the notion
 of $\gamma$-convergence and we show that monotone decreasing costs which are
 $\g$-lower semicontinuous always admit a minimizer among domains with given
 measure contained in a bounded design region. This will imply, as a
 consequence, the existence of a solution for problem (\ref{ar11})  under 
monotonicity assumptions on the function $\Phi$. 
}
\item{ In section \ref{ar3} the case of geometrical constraints on the class
  of admissible domains is considered. Starting from the very strict case of
 admissible domains with a Lipschitz constant a priori bounded, some weaker
 conditions are listed, all implying the existence of solutions of shape 
optimization
 problems, and in particular of (\ref{ar11}) for any continuous function 
$\Phi$.
}

\item{ In section \ref{ar4} we consider the very special case of a function
 $\Phi$ which depends only on the first two eigenvalues $\lb_1, \lb_2$. We 
 show that  without any geometrical constraint on the admissible  domains, in 
 this case we obtain the existence of minimizers for problem (\ref{ar11}) for
 any continuous  (even nonmonotone) function $\Phi$.
}
\item{ Finally, section \ref{ar5} is  devoted to some open questions.
}
\end{itemize}

\section{Monotone cost functions}\label{ar2}
In this section we consider general shape optimization problems of the form 
\begin{equation}\label{ar21}
\min \{F(\Om) : |\Om|=m, \Om \in {\cal A}\}
\end{equation}
where the admissible class $\cal A$ is  the family of all quasi-open
 subsets  $\Om$ of a given bounded design region $D$ of $\RR^N$. The choice 
of  quasi-open sets in this framework seems to be very natural (see Buttazzo
 and Dal Maso \cite{bdaer}); we refer to the book \cite{heinonen} for any 
further
detail about these domains which are very useful in potential theory.

To every $\Om \in {\cal A}$ we can associate the function $w_\Om$ which is 
defined as the unique solution of the problem formally written as
\begin{equation}\label{ar22}
\left  \{ 
     \begin{array}{rrl}
        -\Delta  u =  1 &{\rm in} & \Om  \\
        u =0&{\rm on} &\ov D \setminus \Om .
     \end{array}
\right .
\end{equation}
More precisely, $w_\Om$ can be seen as the unique minimizer of  the functional
 defined on $H^1_0(D)$ by
\begin{equation}
F_\Om(u) = \left  \{ 
     \begin{array}{ll}
        \int_D{1\over 2} |\nabla u|^2 dx -\int_D u dx  &{\rm if} \; u=0\; {\rm 
q.e.} \;

{\rm on}\; \ov D \setminus \Om  \\
        + \infty &{\rm otherwise}. 
     \end{array}
\right .
\end{equation}
By the help of the functions $w_\Om$ we can introduce now a convergence on
 the class ${\cal A}$. 
\begin{defn} \label{ard21}
We say that a sequence $(\Om_n)_n$ in $\ca$ $\g$-converges to a domain
 $\Om \in \ca$ if the solutions $w_{\Om_n}$ converge weakly in $H^1_0(D)$ to
 the solution $w_\Om$.
\end{defn}

 \begin{rem}\label{ard22}
{\rm
The convergence  above turns out to be equivalent  (see Sverak \cite{sverak})
 to the $\Gamma$-convergence of the functionals
\begin{equation}
F_{\Om_n}(u) = \left  \{ 
     \begin{array}{ll}
        \int_D{1\over 2} |\nabla u|^2 dx   &{\rm if} \; u \in H^1_0(\Om_n)  \\
        + \infty&{\rm otherwise} 
     \end{array}
\right .
\end{equation}
to the functional $F_\Om$ defined in  the similar way, with respect to the
 $L^2(D)$ strong convergence; this explains why we adopt the notation
 $\g$-convergence.
}
\end{rem}

\begin{rem}
{\rm
The $\g$-convergence  on $\ca$ is not compact; indeed several shape
 optimization problems of the form (\ref{ar21}) do not admit any solution,
 and the introduction of relaxed formulation is needed in order to describe 
 the behavior of minimizing sequences. We do not want to deal here with this
 topic, and we refer the interested reader to the paper by Buttazzo and Dal
 Maso \cite{bdrelax} where all details about relaxed shape optimization 
problems with Dirichlet boundary conditions can be found.
}
\end{rem}

Even if in general problem (\ref{ar21}) does not admit a solution in $\ca$,
 some particular cases of existence results are available. This is indeed the
 case when the cost $F$ is monotone decreasing with respect to the set
 inclusion. More precisely, the following result holds.

\begin{thm}\label{ard24}
Let $F:\ca \ra [0, +\infty]$ be a functional such that 

\noindent (i) $F$ is $\g$-lower semicontinuous,

\noindent (ii) $F$ is monotone decreasing with respect to the set inclusion.

\noindent Then for every $0\leq m\leq |D|$ problem (\ref{ar21}) admits at
 least a solution in $\ca$.
\end{thm}
The proof of theorem \ref{ard24} was first given by Buttazzo and Dal Maso in
 \cite{bdaer}. Another simpler  proof can be given by using the new concept 
of weak $\g$-convergence, introduced by Bucur, Buttazzo and  Henrot  in 
\cite{bbh}.  In this way, the existence proof follows by the direct methods of
 the calculus of variations.

We want to remark that assumption (i) in theorem \ref{ard24} is verified for
 a large number of situations; for instance we have (see Buttazzo and Dal
 Maso \cite{bdaer}) 
\[ \Om _n \sr{\g}{\lra} \Om\;\;\;\;\;\; \Longrightarrow \;\;\;\;\;\; \lb_j
 (\Om_n) \ra \lb_j (\Om) \;\;\;{\rm for}\; {\rm all}\;\; j \in \NN.\]
Therefore the cost functions
\[F(\Om)=\Phi(\lb(\Om))\]
satisfies assumption (i) as soon as the function $\Phi : \RR^{\NN} \ra [0,
 + \infty]$ is lower semicontinuous, in the sense that
\begin{equation}\label{ar23}
z_j^n \ra z_j  \;\;{\rm for}\; {\rm all}\;\; j \in \NN \;\; \Longrightarrow
 \;\;\Phi (z) \leq \liminf_\nif  \Phi (z^n).
\end{equation}
Assumption (ii) is on the contrary much more restrictive. However, in 
the case of problems involving eigenvalues, of the form (\ref{ar11}), 
theorem \ref{ard24} includes the case of functions $\Phi$ which are monotone
 increasing, that is
\begin{equation}\label{ar24}
z_j^1 \leq z_j^2 \;\;{\rm for}\; {\rm all}\;\; j \in \NN \;\; \Longrightarrow
 \;\; \Phi (z^1)\leq \Phi (z^2).
\end{equation}
This relies on the well known fact that all the eigenvalues of an elliptic
 operator with Dirichlet boundary conditions are decreasing functions of
 the domain. Therefore, summarizing, from theorem \ref{ard24}  we can 
deduce the following result.
\begin{cor}\label{ard25}
Let $\Phi: \RR^N \ra [0, +\infty]$ be a function which is lower 
semicontinuous in the sense of (\ref{ar23}), and monotone decreasing in the 
sense of
 (\ref{ar24}).
Then the optimization problem
\[\min \{\Phi (\lb(\Om)) : |\Om |=m, \Om \in \ca\}\]
admits at least a solution in $\ca$.
\end{cor}




\section{Geometrical constraints}\label{ar3}
If the functional $F$ is not decreasing with respect to the set inclusion, one 
has to impose
 some geometrical constraints on the admissible domains, in order to obtain
 the $\g$-convergence of a minimizing sequence to a domain, in other words to 
obtain the $\g$-compactness of minimizing sequences. This will be done by
 endowing the admissible class $\ca$ with a suitable topology $\tau$ and by 
imposing to elements of $\ca$ suitable geometrical constraints. The role of
 these constraints is double : on one side they make the topological space 
$(\ca, \tau)$ compact, and on the other side they provide  the
 $\g$-convergence of any $\tau$-convergent sequence.



Some good tools to deal with this problem are the so called Mosco conditions, 
which are equivalent to the $\g$-convergence for a
 sequence of open sets. Suppose that $(\Om_n)_n $ is a sequence of open sets
 contained in a bounded design region $D$. It is said that the sequence of 
spaces  $H^1_0(\Om_n)$
converges in the sense of Mosco to the space $H^1_0(\Om)$ if the following 
conditions
are satisfied:
\begin{enumerate}
\item{For all $\phi  \in H^1_0(\Om)$ there exists a sequence $\phi_n 
\in H^1_0(\Om_n)$ such that $\phi_n $ converges strongly in $H^1_0(D)$ to
 $\phi$.}
\item{For any sequence $\phi_{n_k} \in H^1_0(\Om_{n_k})$ weakly convergent
 in $H^1_0(D)$ to a function $\phi$ we have $\phi \in H^1_0(\Om)$.}
\end{enumerate}
If we choose as $\tau$ the  Hausdorff complementary topology on $\ca $, given 
by the metric
\[d_{H^c}(\Om_1, \Om_2)=\sup_{x \in \RR^N}|d(x, \Om_1^c)-d(x, \Om_2^c)|,\]
then it is easy to see that the first Mosco condition is fulfilled for every 
sequence $\Om_n \sr{\tau}{\lra}\Om$. Moreover, as it is well known, the space 
$(\ca, \tau)$ is compact, even if it does not turn out to be $\g$-compact.


In order to satisfy the second Mosco condition, and not to lose compactness, 
one has to impose some  geometrical constraints. We recall now some classical 
results having these properties. The following classes are $\g$-compact, 
listed in an order going from the strongest constraints to the weakest ones.
\begin{itemize}
\item{The class of convex sets contained in $D$.}
\item{The class of Lipschitz  subsets of $D$ with the Lipschitz constant a 
priori bounded (see \cite{pironneau}, \cite{chenais}).}
 
\item{The class of domains satisfying a uniform exterior cone property, i.e. 
such that for every point $x_0$ on the boundary of any $\Om \in \ca$ there is 
a cone, with uniform height and opening, and with vertex in $x_0$, lying in 
the complement of $\Om$.}
\item{The class of domains satisfying a uniform flat cone condition (see 
Bucur, Zol\'esio \cite{ndimjde}), i.e. as above, but with the weaker 
requirement that the cone may be flat, that is of dimension $N-1$.}
\item{The class of domains  satisfying a uniform capacitary density condition, 
i.e. such that there exist $c,r>0$ such that for any $\Om\in \ca$, and for any 
$x \in \partial \Om$, we have
\[ \forall t \in (0,r)\;\;\;\; {{\cp}(\Om^c \cap B_{x,t}, B_{x,2t}) \over 
{\cp}( B_{x,t}, B_{x,2t})}\geq c,\]
where $B_{x,s}$ denotes the ball of radius $s$ centered at $x$.
}

\item{The class of domains  satisfying a uniform Wiener condition, i.e. 
domains satisfying for any $\Om \in \ca$ and for any point  $x \in \partial 
\Om$
\[ 
 \int _r^R {{\cp} (\Omega ^c \cap  {
B}_{x,t}, B_{x,2t}) \over {\cp} (  { B}_{x,t}, B_{x,2t})}{ dt \over
t}\geq g(r,R,x)\; \; \hbox {for every}\;\; 0<r<R<1 \]
where 
$g:(0,1)\times (0,1)\times B \rightarrow \RR_+$ is fixed, such that
for every $ R \in (0,1)$ and every $x\in B$, $ \lim_{r\rightarrow 0} g(r,
R, x) =+\infty$ locally uniformly  on $x$.
}
\end{itemize}
Another interesting $\g$-compact class which is only of topological type,  was 
given by Sverak \cite{sverak} and consists in the following.
\begin{itemize}
\item{For $N=2$, the class of all open subsets $\Om$ of  $D$ for which the 
number of connected components of $\ov D \setminus \Om$ is uniformly bounded.}
\end{itemize}
In fact, it can be shown (see Bucur \cite{bucur2.1}) that this result is a 
consequence of the capacitary density condition, and in $N$ dimensions it was 
generalized by Bucur and Trebeschi \cite{buctreb} for the $p$-Laplacian, with 
$p \in (N-1, N]$.
























\section{The first two eigenvalues}\label{ar4}

In this section we consider the very special case of minimum problems 
involving only the first two eigenvalues, that is
\begin{equation}\label{ar41}
\min\{ \Phi (\lb_1(\Om), \lb_2(\Om)) : |\Om| \leq m, \Om \in {\ca} \}.
\end{equation}
As in the previous situations we take the a priori bounded case
\[\ca =\{ \Om\;\; \hbox{quasi-open}, \; \Om \sq D\}\]
being $D$ a given bounded open subset of $\RR^N$. We assume for simplicity
 that $D$ is a large ball or at least that $D$ contains a sufficiently large
 ball.

The structure of the set $E$ of all attainable  points of $\RR^2$ whose first 
and second coordinates are respectively the first  and second eigenvalues  of 
$-\Delta$  on $H^1_0(\Om)$, with $|\Om| \leq m$ and $\Om \in \ca$, has been 
recently studied by Bucur, Buttazzo and Figueiredo \cite{isabel}. Using a 
result by Ashbaugh and Benguria \cite{ashbaugh} it is possible to show that

\begin{itemize}
\item{ the set $E$ is bounded by the lines 
\[x=\lb_1(B),\;\;\;  y=\lb_1(A),\;\;\;  x=y,\;\;\; y=x{\lb_2(B) \over 
\lb_1(A)}\]
being $B$ the ball of measure $m$ and $A$ the union of two disjoint balls of 
measure ${m \over 2}$ each.
}
\end{itemize}
In the paper \cite{isabel} we prove that 
\begin{itemize}
\item{the set $E$ is closed in $\RR^2$.
}
\end{itemize}
As a consequence we obtain the following existence result.

\begin{thm} \label{ard41}
Let $\Phi : \RR^2 \ra [0, +\infty]$ be any lower semicontinuous function, and 
define
\begin{equation}\label{ar42}
\Phi ( +\infty, +\infty) =\liminf_{x,y \ra \infty} \Phi (x, y).
\end{equation}
Then the optimization problem (\ref{ar41}) admits at least a solution.
\end{thm}
\begin{proof}
Take a minimizing sequence $(\Om_n)$; if $\lb_2(\Om_n) \ra +\infty$, due to
 the structure of the set $E$ we also have $\lb_1(\Om_n) \ra + \infty$.
 Therefore, if for the empty set we define $\lb_1 (\emptyset) =\lb_2
(\emptyset)=+\infty$, thanks to the position (\ref{ar42}) we may conclude
 that the empty set is a minimizer for problem (\ref{ar41}). 

Assume on the contrary that $\lb_2(\Om_n)$ is bounded. The existence of a
 minimizer for problem (\ref{ar41}) then follows straightforward,  since the
 set $E$ is closed and the function $\Phi$ is lower semicontinuous.
\end{proof}

\begin{rem}
{\rm An interesting question is to see if the subset $E$ of $\RR^2$ is convex. 
When the dimension $N=2$ there is a numerical evidence of this fact (see 
Keller and Wolf \cite{wolf}) but up to now we do not know any proof or 
counterexample.
}
\end{rem}

Several other open problems can be raised, in relation to the quite 
surprising result of theorem \ref{ard41}. Some of them are listed  below:
\begin{enumerate}
\item{Is the set $E$ still closed if instead of $\lb_1$ and $\lb_2$ we
 consider $\lb_j$ and $\lb_k$, or more generally $(\lb_j)_{j \in K}$ where
 $K$ is a given set of integer numbers? In particular, if $K = \NN$, is
 the set $E$ of all sequences $\lb(\Om)$ with $\Om \in \ca$ and $|\Om| \leq m$
 closed in $\RR^{\NN}$?
}
\item{Is the set $E$ still closed if instead of the Laplace operator $-\Delta$
 we consider an elliptic operator of the form 
$- {\rm div}  (a(x) \nabla u)$
with the matrix $a(x)$ bounded and uniformly elliptic?
}
\item{ Do points on $\partial E$ correspond to smooth domains $\Om$?
}
\end{enumerate}
Some more numerical  computations about problems involving the first two 
eigenvalues can be found in Finzi Vita \cite{fv96}.

\section{Further remarks and open problems}\label{ar5}

According to  theorem \ref{ard24} of section \ref{ar2}, the existence of an 
optimal domain minimizing the functional $\Om \mapsto F (\Om)$ is obtained for
 quasi-open sets $\Om \sq D$ with $|\Om| =m$, where $D$ is a bounded design
 region. It is natural to think that the minimizing set depends on  $D$. 

Two natural questions can be formulated.
\begin{problem}\label{arp1} 
If by $\ca$ we denote the family of all bounded quasi-open sets of $\RR^N$ 
does problem (\ref{ar21}) still have a solution?
\end{problem}
\begin{problem}\label{arp2}
If a minimizer exists, is this set open and smooth?
\end{problem}
Only for few functionals $F$ these problems were solved. This is the case of 
$\lb_1, \lb_2$, but in general, for  functionals satisfying the assumptions of 
theorem \ref{ard24}, the answer is negative.

Consider for example the divergence operator
\[ Au=-{\rm div} (a(x) \nabla u)\]
where $a : \RR^N \ra \RR$ is given by  $a(x)= {2+ |x| \over 1+ |x|}$. Denoting 
 by $F(\Om)$ the first eigenvalue of $A$ on $\Om$, one can easily remark that 
$F$ satisfies the assumptions of theorem \ref{ard24} and that
\[ \inf \{F(\Om) : |\Om| =m, \Om \in \ca\}=  \lb_1(B)\]
where $\lb_1(B)$ is the first eigenvalue of the Laplacian on the ball of
 measure $m$. A simple computation proves that this infimum is not attained,
 and hence problem \ref{arp1} does not have in general a solution if we drop 
the assumption that admissible domains have to be contained in a bounded 
design region $D$.

For a counterexample to problem \ref{arp2} it is sufficient to consider a 
bounded
 quasi-open quasi-connected set $A$ which is not open, and with measure
 equal to $m$. Setting 
$F(\Om)=\lb _1(\Om\cap A)$ where $\lb _1(\Om\cap A)$ is the first eigenvalue
 of the Laplacian $-\Delta$ on $\Om\cap A$, we have that $F$ satisfies the 
assumptions of theorem \ref{ard24} and has the unique (up to zero capacity 
sets) minimizer $\Om = A$. Hence a minimizer exists, but it is not open.

However, for minimization problems of functions of eigenvalues, of the form 
(\ref{ar11}) with $D=\RR^N$ we do not know an answer to problems 1 and 2, even 
if $\Phi$ is monotone in the sense of (\ref{ar24}).



For some particular functionals  $F$, a slightly different version of problem
 \ref{arp1} is solved in \cite{buhe98}. It is quite natural to extend the
 family of admissible domains to quasi-open sets of $\RR^N$ of measure equal
 to $m$, but which are not necessarily bounded. Following \cite{willem}, the
 compact injection $H^1_0(\Om) \hr  L^2(\Om)$ holds, and hence the spectrum
 of the Laplacian consists of eigenvalues. For example, in \cite{buhe98}
 Bucur and Henrot prove the existence of a minimizer for $\lb_3$ in this 
class. The key tool is a concentration-compactness theorem  in relation with 
the $\g$-convergence (see \cite{bu98cc}).
 In fact, one can prove that if a minimizer in $\ca$ exists  
 and is bounded  for $\lb_h$ with $h=1,2,...,k$, then a 
 minimizer exists for $\lb_{k+1}$. We refer to \cite{bu98cc}, \cite{buhe98}
 for further details. 

The concentration-compactness principle for the $\gamma$-convergence, can
 solve  the minimization problem (\ref{ar41}) if the family of admissible
 domains is extended to all quasi-open sets of $\RR^N$ with measure less than
or equal to $m$ (see \cite{bu98cc}). 

Several other questions about the minimization of some particular functionals 
of
eigenvalues are discussed in a recent paper by Ashbaugh \cite{as98}.


A different question, more related to the regularity of minimizers is the
 following one. When minimizing $\lb_1$ in the family of quasi-open sets of
 fixed measure contained in a bounded design region $D$, then theorem 
\ref{ard24}
 proves the existence of a minimizer. It is obvious that if the design region 
$D$
is large ``enough'', to contain  a ball $B$  of measure $m$, then
 the minimizer is $B$. If $D$ does not contain such a ball, we can prove (by 
using a quite involved argument which relies on results by Brock 
\cite{brockhab} and Hayouni \cite{ha98})  that every minimizer must touch the 
boundary. When the minimizer $\Om$ is smooth and simply connected, a more 
direct procedure is to argue by contradiction and to perform the shape 
derivative which gives
\begin{equation}
 \left  \{ 
     \begin{array}{rcl}
        -\Delta u  &= & \lb_1(\Om) u \;{\rm on} \; \Om \\
        u&=&0\;{\rm on} \; \partial \Om\\
       {\partial u\over \partial n}&=&c\;{\rm on} \; \partial \Om.
     \end{array}
\right .
\end{equation}
This is the Serrin problem and it is known  that $\Om$  must be a ball.
Hence one can formulate the following problems.
\begin{problem}\label{arp3}
Prove (or disprove) the existence of a smooth minimizer $\Om$ in the case $D$ 
smooth, or more generally that $D \cap \partial \Om$  is smooth for every 
(smooth or not) $D$.
\end{problem}
It will be also interesting to prove (or disprove) that in the cases  $\Om$ 
touches the boundary $\partial D$, $\partial \Om$ arrives tangentially on 
$\partial D$.
\begin{problem}\label{arp4}
If a smooth  minimizer exists, prove (or disprove) that all connected 
components of
 $ D \cap \partial \Om$ have constant curvature.
\end{problem} 
\begin{problem}\label{arp5}
If problem \ref{arp4} has a positive answer, prove (or disprove) that the 
constant is the same for
 each connected component.
\end{problem}

In fact, one can perform a local shape derivative on each connected
 component of $ D \cap \partial \Om$ and obtain
 ${\partial u \over \partial n}=$ constant, and the question then becomes a
 ``local'' version of the Serrin problem. 

Other questions which are related to different versions of the Serrin problem 
are the following.

\begin{problem}\label{arp6}
Is the minimizer simply connected if we assume $D$ simply connected?
\end{problem}
\begin{problem}\label{arp7}
Is the minimizer convex if we assume $D$ convex?
\end{problem}
Note that  the regularity problem \ref{arp3} can be raised for every
 minimization problem of functions of eigenvalues, like (\ref{ar11}).

An interesting question is to find a ``continuous'' deformation of an 
arbitrary open set into a ball of same measure, such that the first 
eigenvalue is decreasing and the measure is kept constant. The continuity
 here might be of different types: continuity of the second eigenvalue,
 continuity of all the spectrum, $\g$-continuity, etc. It is not difficult
 to see  that the results  on the set $E$ obtained in \cite{isabel} give 
 the existence of a deformation $t \mapsto  \Om_t$, $0\leq t \leq 1$,  
satisfying the following
 conditions:

\begin{enumerate}
\item{$\Om_0=\Om$, $\Om_1=B$;}
\item{$|\Om_t| = m$;}
\item{$ t \mapsto \lb_1(\Om_t)$ is continuous and not increasing;}
\item{$ t \mapsto \lb_2(\Om_t)$ is continuous.}
\end{enumerate}



A more difficult question, is to find a transformation  satisfying properties
 1), 2),3) above  and which is $\g$-continuous. Of course, such a 
transformation
 satisfies  condition 4). A partial answer was given by Bucur and Henrot
 in \cite{bucur2}, where only the $\g$-continuity on the left was proved.
 Nevertheless, this result seems to be true, and the proof will appear in a
 subsequent article. The main tool is the fine study of the continuous Steiner
 symmetrization introduced in Brock \cite{brock95}.




\vskip .7 cm
{\bf Acknowledgements.}
The second author gratefully acknowledges the hospitality of Laboratoire de 
Math\'ematiques of Universit\'e de Besan\c con, where this paper has been 
written.

\vskip .7 cm
















 
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\end{document}