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: λ 1 ( Ω ∗ ) = min | Ω | = a Ω ⊂ D λ 1 ( Ω ) , where λ 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 Ω ∗ in any case and in any dimension. Full regularity is obtained in dimension 2.