Mathematics

In this talk I will discuss the maximization of the first eigenvalue of the Laplacian (or an isotropic elliptic operator with nonconstant coefficients) with Dirichlet boundary conditions on a domain of the kind Ω\Σ, where Ω is a given open set and Σ is an unknown compact set.The class of admissible Dirichlet regions among which I look for the optimum are different, according to the space dimension. In one dimension, when Ω is an interval, Σ is a discrete set of n points, with fixed n. In two dimensions, when Ω is a planar domain, I require Σ to be connected set with prescribed lenght L.I will focus on some qualitative results on such maximizers and then, by means of Γ-convergence theory, on the asymptotics as n → ∞ or L → ∞. I will also discuss a comparison with the maximum distance problems. The results are joint work with Paolo Tilli (Politecnico di Torino).