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

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.