*Published Paper*

**Inserted:** 26 may 2016

**Last Updated:** 26 may 2016

**Journal:** Applicable Analysis

**Volume:** 90

**Number:** 2

**Pages:** 263 - 278

**Year:** 2011

**Abstract:**

In this paper, we consider the well-known following shape optimization problem:
\[\lambda_2(\Omega^*)=\min_{\stackrel{

\Omega

=V_0} {\Omega\textrm{ convex}}} \lambda_2(\Omega),\]
where $\lambda_2(\Omega)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\Omega\subset\mathbb{R}^2$, and $

\Omega

$ is the area of $\Omega$. We prove, under some technical assumptions, that any optimal shape $\Omega^*$ is $C^{1,\frac{1}{2}}$ and is not $C^{1,\alpha}$ for any $\alpha>\frac{1}{2}$. We also derive from our strategy some more general
regularity results, in the framework of partially overdetermined boundary value problems, and we apply these results to some other shape optimization problems.

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