Speaker: Jefferson Abrantes

Universidade Federal de Campina Grande, Brazil

Date:  April 26, 2022 at 12 Santiago time

Abstract: An optimization problem with volume constraint involving the \(\Phi\)-Laplacian in Orlicz-Sobolev spaces is considered for the case where \(\Phi\) does not satisfy the natural condition introduced by Lieberman. A minimizer \(u_\Phi\) having non-degeneracy at the free boundary is proved to exist and some important consequences are established like the Lipschitz regularity of \(u_\Phi\) along the free boundary, that the set \(\{u_\Phi >0\}\) 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 \(\ell \to 0^+\), 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 \(u_\Phi\).

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Gabrielle Nornberg