Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle

We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains Ω having prescribed volume and contained in a fixed box D ; equivalently, we look for the best way to remove a compact set (obstacle) K ⊂ D ‾ of Lebesgue measure | K | = ε , 0 < ε < | D | , in order to minimize the first Dirichlet eigenvalue of the set Ω = D ∖ K . In the small volume regime ε → 0 , we prove that the optimal obstacles accumulate, in a suitable sense, to points of ∂ D where | ∇ ϕ 0 | is minimal, where ϕ 0 denotes the first eigenfunction of the Dirichlet Laplacian on D . Moreover, we provide a fairly detailed description of the convergence of the optimal eigenvalues, eigenfunctions and free boundaries. Our results are based on sharp estimates of the optimal eigenvalues, in terms of a suitable notion of relative capacity.