Optimal Shape for Elliptic Problems with Random Perturbations

In this paper we analyze the relaxed form of a shape optimization problem with state equation $\{{array}{ll} -div \big(a(x)Du\big)=f\qquad\hbox{in}D \hbox{boundary conditions on}\partial D. {array}.$ 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 $α\le a\leβ$ and $\displaystyle\int_D a(x) dx\le m$, which minimizes a cost function of the form $\int_Ω\int_Dj\big(x,ω,u_a(x,ω)\big) dx dP(ω).$ 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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