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About Hölder-regularity of the convex shape minimizing λ2

In this paper, we consider the well-known following shape optimization problem: $$λ_2(Ω^*)=\min_{\stackrel{|Ω|=V_0} {Ω\textrm{ convex}}} λ_2(Ω),$$ where $λ_2(\Om)$ 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 $Ω^*$ is $\mathcal{C}^{1,\frac{1}{2}}$ and is not $\C^{1,α}$ for any $α>\frac{1}{2}$. 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.