Optimal design problems for a degenerate operator in Orlicz–Sobolev spaces

Abstract

An optimization problem with volume constraint involving the $$\varPhi $$ -Laplacian in Orlicz–Sobolev spaces is considered for the case where $$\varPhi $$ does not satisfy the natural condition introduced by Lieberman. A minimizer $$u_\varPhi $$ having non-degeneracy at the free boundary is proved to exist and some important consequences are established like the Lipschitz regularity of $$u_ \varPhi $$ along the free boundary, that the set $$\{u_\varPhi >0\}$$ has uniform positive density, that the free boundary is porous with porosity $$\delta >0$$ and has finite $$(N-\delta )$$ -Hausdorff measure. Under a geometric compatibility condition set up by Rossi and Teixeira, it is established the behavior of a $$\ell $$ -quasilinear optimal design problem with volume constraint for $$\ell $$ small. As $$\ell \rightarrow 0^+$$ , we obtain a limiting free boundary problem driven by the infinity-Laplacian operator and find the optimal shape for the limiting problem. The proof is based on a penalization technique and a truncated minimization problem in terms of the Taylor polynomial of $$\varPhi $$ .