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 Omega having prescribed volume and contained in a fixed box D; equivalently, we look for the best way to remove a compact set (obstacle) K subset of D of Lebesgue measure K = epsilon, 0 < epsilon < D , in order to minimize the first Dirichlet eigenvalue of the set Omega = D \ K. In the small volume regime epsilon -> 0, we prove that the optimal obstacles accumulate, in a suitable sense, to points of partial derivative D where del|phi(0)| is minimal, where phi(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. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).