A limiting free boundary problem for a degenerate operator in Orlicz-Sobolev spaces
Summary: A free boundary optimization problem involving the \(\Phi \)-Laplacian in Orlicz-Sobolev spaces is considered for the case where \(\Phi\) does not satisfy the natural conditions introduced by Lieberman. A minimizer \(u_\Phi\) having non-degeneracy at the free boundary is proved to exist and some important consequences are established, namely, the Lipschitz regularity of \(u_\Phi\) along the free boundary, that the positivity set of \(u_\Phi\) has locally uniform positive density, and that the free boundary is porous with porosity \(\delta > 0\) and has finite \((N - \delta)\)-Hausdorff measure. The method is based on a truncated minimization problem in terms of the Taylor polynomial of \(\Phi\) of order \(2k\). The proof demands to revisit the Lieberman proof of a Harnack inequality and verify that the associated constant with this inequality is independent of \(k\) provided that \(k\) is sufficiently large.
scientific article; zbMATH DE number 7318498
scientific article