JournalsrmiVol. 36, No. 6pp. 1687–1720

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

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Abstract

A free boundary optimization problem involving the Φ\Phi-Laplacian in Orlicz–Sobolev spaces is considered for the case where Φ\Phi does not satisfy the natural conditions introduced by Lieberman. A minimizer uΦu\Phi having non-degeneracy at the free boundary is proved to exist and some important consequences are established, namely, the Lipschitz regularity of uΦu\Phi along the free boundary, that the positivity set of uΦu\Phi has locally uniform positive density, and that the free boundary is porous with porosity δ>0\delta > 0 and has finite (Nδ)(N − \delta)-Hausdorff measure. The method is based on a truncated minimization problem in terms of the Taylor polynomial of Φ\Phi of order 2k2k. The proof demands to revisit the Lieberman proof of a Harnack inequality and verify that the associated constant with this inequality is independent of kk provided that kk 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