Square function/non-tangential maximal function estimates and the dirichlet problem for non-symmetric elliptic operators

Steve Hofmann, Carlos Kenig, Svitlana Mayboroda, Jill Pipher

Research output: Contribution to journalArticlepeer-review

66 Scopus citations

Abstract

We consider divergence form elliptic operators (formula presented), defined in the half space (formula presented), where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation Lu= 0, and we then combine these estimates with the method of “ϵ-approximability” to show that L-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class Awith respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in Lp, for some p < ∞). Previously, these results had been known only in the case n = 1.

Original languageEnglish (US)
Pages (from-to)483-529
Number of pages47
JournalJournal of the American Mathematical Society
Volume28
Issue number2
DOIs
StatePublished - Apr 1 2015

Keywords

  • AMuckenhoupt weights
  • Dirichlet problem
  • Divergence form elliptic equations
  • Harmonic measure
  • Layer potentials
  • Non-tangential maximal function
  • Square function
  • ∊-approximability

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