Shape optimization problems in control form

TL;DR

This paper investigates shape optimization problems involving the p-Laplacian operator, establishing conditions for the existence of optimal domains, which are shown to have finite perimeter and be open sets under certain assumptions.

Contribution

It provides new existence results for optimal domains in shape optimization problems with the p-Laplacian, highlighting differences between cases p>d and p≤d.

Findings

Optimal domains have finite perimeter.

Under certain conditions, optimal domains are open sets.

Existence results vary significantly between p>d and p≤d cases.

Abstract

We consider a shape optimization problem written in the optimal control form: the governing operator is the $p$-Laplacian in the Euclidean space $\R^d$, the cost is of an integral type, and the control variable is the domain of the state equation. Conditions that guarantee the existence of an optimal domain will be discussed in various situations. It is proved that the optimal domains have a finite perimeter and, under some suitable assumptions, that they are open sets. A crucial difference is between the case $p>d$, where the existence occurs under very mild conditions, and the case $p\le d$, where additional assumptions have to be made on the data.