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: lambda(1)(Omega*) = min lambda(1) (Omega), |Omega|=u Omega subset of D 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. (C) 2008 Elsevier Masson SAS. All rights reserved.