Optimal shape for elliptic problems with random perturbations

Giuseppe Buttazzo; Faustino Maestre

doi:10.3934/dcds.2011.31.1115

Article Contents

2011, Volume 31, Issue 4: 1115-1128. Doi: 10.3934/dcds.2011.31.1115

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Optimal shape for elliptic problems with random perturbations

Giuseppe Buttazzo^1, and
Faustino Maestre^2,

1.
Dipartimento di Matematica, Largo B. Pontecorvo, 5, 56127 Pisa
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Tarﬁa s/n, E-41012 Sevilla

Received: September 30, 2009

Revised: February 28, 2010

Published: September 2011

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Abstract

In this paper we analyze the relaxed form of a shape optimization problem with state equation $$ \begin{equation} \left\{\begin{array}{ll} -div\big(a(x)Du\big)=f\qquad\hbox{in }D\\ \hbox{boundary conditions on }\partial D. \end{array} \right. \end{equation} $$ 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, which evidence the previous analytical results.

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Mathematics Subject Classification: Primary: 49J20, 49K30, 49K45.

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