About Hölder-regularity of the convex shape minimizing λ2

Abstract

In this paper, we consider the well-known following shape optimization problem: $${\lambda }_2({\Omega }^{\ast })=\underset{\stackrel{|\Omega |=V_0}{\Omega \text{ convex}}}{\min }{\lambda }_2(\Omega ),$$ 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 ${\Omega }^{\ast }$ is ${\mathcal{C}}^{1,\frac{1}{2}}$ and is not $\C^{1,α}$ 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.