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

We consider the well-known following shape optimization problem: λ1(Ω ∗) = min |Ω|=a Ω⊂D λ1(Ω), where λ1 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 theo-rem). 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