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# Optimal Shape for Elliptic Problems with Random Perturbations

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### Abstract

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 $$\xi(x,\omega)$$. 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.

### Most cited references8

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### Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion

(1984)
This paper is a sequel to [1-4]. We consider the problem of G-closure, i.e. the description of the setGUof effective tensors of conductivity for all possible mixtures assembled from a number of initially given components belonging to some fixed setU. Effective tensors are determined here in a sense of G-convergence relative to the operator ∇·D· ∇, of the elementsDeU∈ [5, 6].
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### H-Convergence

(1997)
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### Variational Methods for Structural Optimization

(2000)
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### Author and article information

###### Journal
1002.2770

Analysis, Numerical methods