An optimal design problem with non-standard growth and no concentration effects

TL;DR

This paper derives an integral representation for functionals in optimal design and damage evolution problems with non-standard growth, including perimeter penalization, and explores dimension reduction for thin film design.

Contribution

It provides a new integral representation for these functionals under non-standard growth conditions, without singular terms, and addresses dimension reduction in thin film optimal design.

Findings

Integral representation includes absolutely continuous and perimeter terms.

No additional singular term in the representation.

Results applicable to optimal design of thin films.

Abstract

We obtain an integral representation for certain functionals arising in the context of optimal design and damage evolution problems under non-standard growth conditions and perimeter penalisation. Under our hypotheses, the integral representation includes a term which is absolutely continuous with respect to the Lebesgue measure and a perimeter term, but no additional singular term. We also study some dimension reduction problems providing results for the optimal design of thin films.