About Hölder-regularity of the convex shape minimizing λ2

Jimmy Lamboley (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: shape optimization; eigenvalues of the Laplacian; regularity of free boundaries; conformal map; convex constraint; overdetermined boundary value problems

Domaines: Optimisation et contrôle [math.OC]

Jimmy Lamboley. About Hölder-regularity of the convex shape minimizing λ2. Applicable Analysis, 2010, pp.1. ⟨hal-00530272⟩