BMO Solvability and \(A_{\infty }\) Condition of the Elliptic Measures in Uniform Domains

Article

Abstract

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e., \(\omega _L\in A_{\infty }(\sigma )\). This generalizes a previous result on Lipschitz domains by Dindos, Kenig, and Pipher (see Dindos et al. in J Geom Anal 21:78–95, 2011).

Keywords

Harmonic measure Uniform domain \(A_{\infty }\) Muckenhoupt weight BMO solvability Carleson measure 

Mathematics Subject Classification

35J25 31B35 42B37 

Notes

Acknowledgements

The author was partially supported by NSF DMS Grants 1361823 and 1500098. The author wants to thank her advisor Prof. Tatiana Toro for introducing her to this area and the enormous support given by her during the work on this paper. The author also wants to thank Prof. Hart Smith for his support, and thank the referee for the careful reading and helpful suggestions.

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© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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