Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

Steve Hofmann, Phi Le, Andrew Morris

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1 Citation (Scopus)
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Abstract

We prove that the Dirichlet problem for degenerate elliptic equations div(A∇u)=0 in the upper half-space (x,t)∈Rn+1+ is solvable when n≥2 and the boundary data is in Lpμ(Rn) for some p<∞. The coefficient matrix A is only assumed to be measurable, real-valued and t-independent with a degenerate bound and ellipticity controlled by an A2-weight μ. It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure is in A∞ with respect to the μ-weighted Lebesgue measure on Rn. The Carleson measure estimate allows us to avoid applying the method of ϵ-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.
Original languageEnglish
Pages (from-to)2095-2146
Number of pages52
JournalAnalysis and PDE
Volume12
Issue number8
DOIs
Publication statusPublished - 28 Oct 2019

Keywords

  • square functions
  • nontangential maximal functions
  • harmonic measure
  • Radon–Nikodym derivative
  • Carleson measure
  • divergence form elliptic equations
  • Dirichlet problem
  • A2 Muckenhoupt weights
  • reverse Hölder inequality
  • Harmonic measure
  • Radon-Nikodym derivative
  • Square functions
  • Divergence form elliptic equations
  • Nontangential maximal functions
  • Reverse Hölder inequality
  • A Muckenhoupt weights

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Numerical Analysis

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