*Preprint*

**Inserted:** 12 nov 2009

**Year:** 2009

**Abstract:**

In this paper we analyze the relaxed form of a shape optimization problem with state equation $$\left\{\begin{array}{ll} -div\big(a(x)Du\big)=f\qquad\hbox{in }D

\hbox{boundary conditions on }\partial D.
\end{array}\right.$$
The new fact is that the term $f$ is only known up to a random
perturbation $\xi(x,\omega)$. The goal is to find an optimal
coefficient $a(x)$, fulfilling the usual constraints $\alpha\le
a\le\beta$ and $\displaystyle\int_D a(x)\,dx\le m$, which
minimizes a cost function of the form
$$\int_{\Omega\int}_{Dj\big}(x,\omega,u_{a}(x,\omega)\big)\,dx\,dP(\omega).$$
Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.

**Keywords:**
Homogenization, shape optimization, random perturbation, elliptic state equation

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