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

TL;DR

This paper investigates the regularity of optimal shapes minimizing the first Laplacian eigenvalue under volume and inclusion constraints, proving full boundary regularity in two dimensions and general regularity properties in higher dimensions.

Contribution

It establishes regularity properties of optimal shapes in a shape optimization problem, including full regularity in 2D and general boundary regularity in higher dimensions.

Findings

Full boundary regularity in 2D.

Regularity properties of optimal shapes in any dimension.

Abstract

We consider the well-known following shape optimization problem: $$\lambda_1(\Omega^*)=\min_{\stackrel{|\Omega|=a} {\Omega\subset{D}}} \lambda_1(\Omega), $$ 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 $\Omega^*$ in any case and in any dimension. Full regularity is obtained in dimension 2.