Relaxation for an optimal design problem in $BD(\Omega)$

TL;DR

This paper develops a measure-based relaxation approach for an optimal design problem involving functions of bounded deformation and perimeter constraints, providing a new representation of the energy functional.

Contribution

It introduces a measure representation for the relaxed energy functional in optimal design problems with linear growth in the space of bounded deformation.

Findings

Measure representation of the relaxed functional.

Extension of relaxation techniques to $BD(\Omega)$.

Application to problems with perimeter and symmetrised gradient terms.

Abstract

We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(\chi,u)$, where $\chi$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded deformation, of an energy with a bulk term depending on the symmetrised gradient as well as a perimeter term.