The Optimal Design Problem of Céa and Malanowski Revisited

Abstract

In this paper we consider the optimal design problem for the compliance as studied by Céa and Malanowski in 1970. It consists in finding the optimal distribution of two materials in a fixed domain. This type of problem often leads to the occurence of a microstructure or a composite material. We complete the characterization of the solutions and establish the existence of optimal partitions into two measurable subdomains. Moreover we show that the set of solutions of the convex relaxed problem always contains measurable partitions. A numerical example is presented. The results readily extend to higher order elliptic operators and to variational inequalities over closed convex sets.

Supported in part by a Killam fellowship from Canada Council, National Sciences and Engineering Research Council of Canada operating grant A-8730 and infrastructure grant INF-7939, and by a FCAR grant from the Ministère de l’Education du Québec.