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

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.