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A free boundary optimization problem involving the -Laplacian in Orlicz–Sobolev spaces is considered for the case where does not satisfy the natural conditions introduced by Lieberman. A minimizer having non-degeneracy at the free boundary is proved to exist and some important consequences are established, namely, the Lipschitz regularity of along the free boundary, that the positivity set of has locally uniform positive density, and that the free boundary is porous with porosity and has finite -Hausdorff measure. The method is based on a truncated minimization problem in terms of the Taylor polynomial of of order . The proof demands to revisit the Lieberman proof of a Harnack inequality and verify that the associated constant with this inequality is independent of provided that is sufficiently large.
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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