Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints

Abstract

We consider the well-known following shape optimization problem: $${\lambda }_1({\Omega }^{\ast })=\underset{\stackrel{|\Omega |=a}{\Omega \subset D}}{\min }{\lambda }_1(\Omega ),$$ where ${\lambda }_1$ denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition, and $D$ is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume $a$ if such a ball exists in the box $D$ (Faber-Krahn's theorem). In this paper, we prove regularity properties of the boundary of the optimal shapes ${\Omega }^{\ast }$ in any case and in any dimension. Full regularity is obtained in dimension 2.

Cited by in corpus (6)

• Faber-Krahn inequalities in sharp quantitative form
• Lipschitz regularity of the eigenfunctions on optimal domains
• A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results
• Existence and regularity of Faber Krahn minimizers in a Riemannian manifold
• Regularity of shape optimizers for some spectral fractional problems
• A shape optimization problem in cylinders and related overdetermined problems