Relaxation for an optimal design problem with linear growth and perimeter penalization

TL;DR

This paper studies the relaxation and integral representation of an optimal design problem involving linear growth and perimeter penalization, addressing existence issues and complex interactions in the limit energy.

Contribution

It introduces a relaxation framework for a class of optimal design problems with perimeter penalization, extending to more general models and analyzing the limit energy interactions.

Findings

Established integral representation in BV space for the relaxed energy

Addressed existence issues through perimeter penalization

Analyzed interactions in the limit energy for generalized models

Abstract

The paper is devoted to the relaxation and integral representation in the space of functions of bounded variation for an integral energy arising from optimal design problems. The presence of a perimeter penalization is also considered in order to avoid non existence of admissible solutions, besides this leads to an interaction in the limit energy. Also more general models have been taken into account.