Vol. 12, No. 8, 2019

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Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

Steve Hofmann, Phi Le and Andrew J. Morris

Vol. 12 (2019), No. 8, 2095–2146
Abstract

We prove that the Dirichlet problem for degenerate elliptic equations div(Au) = 0 in the upper half-space (x,t) +n+1 is solvable when n 2 and the boundary data is in Lμp(n) 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 n . 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.

Keywords
square functions, nontangential maximal functions, harmonic measure, Radon–Nikodym derivative, Carleson measure, divergence form elliptic equations, Dirichlet problem, $A_2$ Muckenhoupt weights, reverse Hölder inequality
Mathematical Subject Classification 2010
Primary: 35J25, 35J70, 42B20, 42B25
Milestones
Received: 2 May 2018
Revised: 17 October 2018
Accepted: 30 November 2018
Published: 28 October 2019
Authors
Steve Hofmann
Department of Mathematics
University of Missouri
Columbia, MO
United States
Phi Le
Mathematics Department
Syracuse University
Syracuse, NY
United States
Andrew J. Morris
School of Mathematics
University of Birmingham
Birmingham
United Kingdom