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In this paper, we consider the well-known following shape optimizationproblem: $$\lambda_2(\Omega^*)=\min_{\stackrel{|\Omega|=V_0} {\Omega\textrm{convex}}} \lambda_2(\Omega),$$ where $\lambda_2(\Om)$ denotes the secondeigenvalue of the Laplace operator with homogeneous Dirichlet boundaryconditions 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 generalregularity results, in the framework of partially overdetermined boundary valueproblems, and we apply these results to some other shape optimization problems.

About H\"older-regularity of the convex shape minimizing {\lambda}2