Calculus of Variations and Geometric Measure Theory

G. Buttazzo - F. Maestre

Optimal Shape for Elliptic Problems with Random Perturbations

created by buttazzo on 12 Nov 2009



Inserted: 12 nov 2009

Year: 2009


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\intDj\big(x,\omega,ua(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