*Published Paper*

**Inserted:** 21 may 2016

**Last Updated:** 21 may 2016

**Journal:** Annales de l'IHP, Analyse non linĂ©aire

**Volume:** 26

**Number:** 4

**Pages:** 1149-1163

**Year:** 2009

**Abstract:**

We consider the well-known following shape optimization problem:
\[\lambda_1(\Omega^*)=\min_{\stackrel{

\Omega

=a}
{\Omega\subset{ D}}} \lambda_1(\Omega),
\] where $\lambda_1$ denotes the first eigenvalue of the Laplacian with homogeneous Dirichlet boundary condition and $D$ is an open bounded set (a box).
It is well-known that the solution of this problem is the ball of volume $a$ if such a ball exists in the box $D$ (Faber-Krahn's theorem).\\
In this paper, we prove regularity properties of the boundary of the optimal shapes $\Omega^*$ in any case and in any dimension.
Full regularity is obtained in dimension 2.

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