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2019 Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations
Steve Hofmann, Phi Le, Andrew J. Morris
Anal. PDE 12(8): 2095-2146 (2019). DOI: 10.2140/apde.2019.12.2095


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


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Steve Hofmann. Phi Le. Andrew J. Morris. "Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations." Anal. PDE 12 (8) 2095 - 2146, 2019.


Received: 2 May 2018; Revised: 17 October 2018; Accepted: 30 November 2018; Published: 2019
First available in Project Euclid: 14 December 2019

zbMATH: 07143411
MathSciNet: MR4023976
Digital Object Identifier: 10.2140/apde.2019.12.2095

Primary: 35J25 , 35J70 , 42B20 , 42B25

Keywords: $A_2$ Muckenhoupt weights , Carleson measure , Dirichlet problem , divergence form elliptic equations , harmonic measure , nontangential maximal functions , Radon–Nikodym derivative , reverse Hölder inequality , square functions

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.12 • No. 8 • 2019
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