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: $${\lambda }_1({\mathrm{\Omega }}^{\ast })=\underset{}{\min }{\lambda }_1(\mathrm{\Omega }),$$ 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 ${\mathrm{\Omega }}^{\ast }$ in any case and in any dimension. Full regularity is obtained in dimension 2.