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|=aΩ⊂Dλ1(Ω), where λ1λ1 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∗Ω∗ in any case and in any dimension. Full regularity is obtained in dimension 2.