Optimal shapes for general integral functionals

TL;DR

This paper investigates shape optimization problems for integral functionals, proving existence of solutions that are quasi open sets and conditions under which they are open with finite perimeter.

Contribution

It establishes the existence of solutions as quasi open sets and identifies mild conditions for these solutions to be open with finite perimeter.

Findings

Existence of solutions as quasi open sets

Conditions under which solutions are open and have finite perimeter

Counterexamples showing limitations of these conditions

Abstract

We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $\Omega$ that varies over all subdomains of a given bounded domain $D$ of ${\bf R}^d$. We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.