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We obtain an integral representation for certain functionals arising in thecontext of optimal design and damage evolution problems under non-standardgrowth conditions and perimeter penalisation. Under our hypotheses, theintegral representation includes a term which is absolutely continuous withrespect to the Lebesgue measure and a perimeter term, but no additionalsingular term.  We also study some dimension reduction problems providing results for theoptimal design of thin films.

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