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 \arrayll -div (a(x)Du)=finD boundary conditions on∂ D. array. The new fact is that the term f is only known up to a random perturbation (x,ω). The goal is to find an optimal coefficient a(x), fulfilling the usual constraints α aβ and ∫D a(x) dx m, which minimizes a cost function of the form ∫∫Dj(x,ω,ua(x,ω)) dx dP(ω). Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.
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