A minimum problem with free boundary for the p(x) - Laplace operator

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1 Citation (Scopus)


In this paper we consider the problem of minimizing the functional J(u) = ∫Ω(1/p(x)|∇u|p(x) + Q(x)Χ[u≥0])dx. We prove Lipschitz continuity for each minimizer u and establish the nondegeneracy at the free boundary (∂[u ≥ 0]) ∩ Ω and the locally uniform positive density of the sets [u ≥ 0] and [u = 0]. In particular we obtain that the Lebesgue measure of the free boundary is zero.

Original languageEnglish
Pages (from-to)25-61
Number of pages37
JournalAdvanced Nonlinear Studies
Issue number1
Publication statusPublished - 2011
Externally publishedYes


  • Free Boundary
  • Hölder continuity
  • Lipschitz continuity
  • Minimizer
  • P(x)-Laplace Operator
  • Positive Density

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • General Mathematics


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