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

### Jefferson Abrantes Santos

Universidade Federal de Campina Grande, Brazil### Sergio H. Monari Soares

Universidade de São Paulo, São Carlos, Brazil

## Abstract

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.

## Cite this article

Jefferson Abrantes Santos, Sergio H. Monari Soares, A limiting free boundary problem for a degenerate operator in Orlicz–Sobolev spaces. Rev. Mat. Iberoam. 36 (2020), no. 6, pp. 1687–1720

DOI 10.4171/RMI/1180