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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: $$λ_1(Ω^*)=\min_{\stackrel{|Ω|=a} {Ω\subset{D}}} λ_1(Ω), $$ 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.