Optimal shape for elliptic problems with random perturbations

Abstract

In this paper we analyze the relaxed form of a shape optimization problem with state equation − div a(x)Du = f in D boundary conditions on ∂D. The new fact is that the term f is only known up to a random perturbation ξ(x, ω). The goal is to find an optimal coefficient a(x), fulfilling the usual constraints α ≤ a ≤ β and Z D a(x) dx ≤ m, which minimizes a cost function of the form Z Ω Z D j x, ω, ua(x, ω) dxdP(ω). Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.