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 $\{arrayll-div(a(x)Du)=finDboundary\; conditions\; on\partial D.array.$ The new fact is that the term $f$ is only known up to a random perturbation $\xi (x,\omega )$. The goal is to find an optimal coefficient $a(x)$, fulfilling the usual constraints $\alpha \le a\le \beta$ and ${\int }_{D}a(x)dx\le m$, which minimizes a cost function of the form $${\int }_{\Omega }{\int }_{D}j(x,\omega ,u_a(x,\omega ))dxdP(\omega ).$$ Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.

17 pages, 6 figures