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:

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.