Occupational measures and averaged shape optimization

¹ Department of Applied Mathematics, University of Washington, Lewis Hall 202, Seattle, WA 98195-3925, USA.
² Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA.

^* Corresponding author: torres@math.purdue.edu

Received: 21 January 2016
Revised: 20 February 2017
Accepted: 22 February 2017

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.