Optimal design problems for a degenerate operator in Orlicz-Sobolev spaces.

Abstract: An optimization problem with volume constraint involving the Φ-Laplacian in Orlicz-Sobolev spaces is considered for the case where Φ does not satisfy the natural condition introduced by Lieberman. A minimizer uΦ having non-degeneracy at the free boundary is proved to exist and some important consequences are established like the Lipschitz regularity of uΦ along the free boundary, that the set {uΦ>0} has uniform positive density, that the free boundary is porous with porosity δ>0 and has finite (N−δ)-Hausdorff measure. Under a geometric compatibility condition set up by Rossi and Teixeira, it is established the behavior of a ℓ-quasilinear optimal design problem with volume constraint for ℓ small. As ℓ→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 uΦ.

Posted on Apr 25, 2022 in Differential Equations, Seminars