Resumen de A limiting free boundary problem for a degenerate operator in Orlicz–Sobolev spaces

Jefferson Abrantes Santos, Sergio H. Monari Soares

• A free boundary optimization problem involving the Φ-Laplacian in Orlicz–Sobolev spaces is considered for the case where Φ does not satisfy the natural conditions introduced by Lieberman. A minimizer uΦ having non-degeneracy at the free boundary is proved to exist and some important consequences are established, namely, the Lipschitz regularity of uΦ along the free boundary, that the positivity set of uΦ has locally uniform positive density, and that the free boundary is porous with porosity δ>0 and has finite (N−δ)-Hausdorff measure. The method is based on a truncated minimization problem in terms of the Taylor polynomial of Φ 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.