On the regularity of the free boundary for quasilinear obstacle problems

S. Challal, A. Lyaghfouri, J. F. Rodrigues, R. Teymurazyan

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14 Citations (Scopus)

Abstract

We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the p(x)-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth p(x) > 1, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for p(x)-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of at most a countable family of C1 hypersurfaces: (i) by extending directly the finiteness of the (n - 1)-dimensional Hausdorff measure of the free boundary to the case of heterogeneous p-Laplacian type operators with constant p, 1 < p < ∞; (ii) by proving the characteristic function of the coincidence set is of bounded variation in the case of non degenerate or non singular operators with variable power growth p(x) > 1.

Original languageEnglish
Pages (from-to)359-394
Number of pages36
JournalInterfaces and Free Boundaries
Volume16
Issue number3
DOIs
Publication statusPublished - 2014
Externally publishedYes

Keywords

  • Heterogeneous p-Laplacian
  • Obstacle problem
  • Quasi-linear elliptic operators
  • Regularity of the free boundary

ASJC Scopus subject areas

  • Applied Mathematics

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