The Relaxed Formulation of an Optimal Design Problem via Homogenization Theory

Abstract

AbstractWe consider a classical problem in optimal design which involves a finite number of electric or thermic materials defined by their corresponding diffusion matrices. It consists in obtaining a matrix function in a bounded open set \(\varOmega \subset \mathbb R^N\) by placing in each point one of the materials in such way that the corresponding electric potential or temperature minimizes a certain functional. It is well known that this problem has no solution in general and therefore that it is necessary to work with a relaxed formulation. Usually, this is carried out via homogenization theory replacing the above mixtures by the more general ones described in the previous chapter. Here, we obtain and study this relaxed formulation in the case of a functional which can depend nonlinearly on the gradient of the state function. Thus, it is not continuous in general for the weak topology of \(H^1_0(\varOmega )\). More generally we consider functionals which also depend on the proportion of the materials used in the mixture and the current flow or heat flux.KeywordsOptimal designLower semicontinuous envelopeRelaxationNon-sequentially weakly continuous functionalIntegral representation