# Occupational measures and averaged shape optimization

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS（2018）

Abstract

We consider the minimization of averaged shape optimization problems over the class of sets of finite perimeter. We use occupational measures, which are probability measures defined in terms of the reduced boundary of sets of finite perimeter, that allow to transform the minimization into a linear problem on a set of measures. The averaged nature of the problem allows the optimal value to be approximated with sets with unbounded perimeter. In this case, we show that we can also approximate the optimal value with convex polytopes with n + 1 faces shrinking to a point. We derive conditions under which we show the existence of minimizers and we also analyze the appropriate spaces in which to study the problem.

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Key words

Shape optimization,occupational measures,sets of finite perimeter,Cheeger sets

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