Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

Hofmann, Steve; Le, Phi; Morris, Andrew J.

doi:10.2140/apde.2019.12.2095

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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

DOI: 10.2140/apde.2019.12.2095

Abstract

We prove that the Dirichlet problem for degenerate elliptic equations $div (A \nabla u) = 0$ in the upper half-space $(x, t) \in ℝ_{+}^{n + 1}$ is solvable when $n \geq 2$ and the boundary data is in $L_{μ}^{p} (ℝ^{n})$ for some $p < \infty$ . The coefficient matrix $A$ is only assumed to be measurable, real-valued and $t$ -independent with a degenerate bound and ellipticity controlled by an $A_{2}$ -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 $A_{\infty}$ with respect to the $μ$ -weighted Lebesgue measure on $ℝ^{n}$ . 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

Vol. 12, No. 8, 2019