σ-Finiteness of elliptic measures for quasilinear elliptic PDE in space

Murat Akman, John Lewis, Andrew Vogel

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

In this paper we study the Hausdorff dimension of a elliptic measure μf in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of σ-finite n−1 dimensional Hausdorff measure for p>n and the same result holds for p=n with an assumption on the boundary. We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension ≤n−1−δ for p≥n for some δ which depends on various constants including p. The first result generalizes the authors previous work in [3] when the PDE is the p-Laplacian and the second result generalizes the well known theorem of Wolff in [24] when p=2 and n=2.

Original languageEnglish (US)
Pages (from-to)512-557
Number of pages46
JournalAdvances in Mathematics
Volume309
DOIs
StatePublished - Mar 17 2017

Keywords

  • Hausdorff Dimension of a Borel measure
  • Hausdorff dimension
  • Hausdorff measure
  • Quasilinear elliptic PDEs
  • The four-corner Cantor set

ASJC Scopus subject areas

  • Mathematics(all)

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