  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 $\lambda_2(\Om)$ denotes the second
eigenvalue of the Laplace operator with homogeneous Dirichlet boundary
conditions in $\Om\subset\R^2$, and $|\Om|$ is the area of $\Om$. We prove,
under some technical assumptions, that any optimal shape $\Omega^*$ is
$\mathcal{C}^{1,\frac{1}{2}}$ 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.
