#### 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 $div\left(A\nabla u\right)=0$ in the upper half-space $\left(x,t\right)\in {ℝ}_{+}^{n+1}$ is solvable when $n\ge 2$ and the boundary data is in ${L}_{\mu }^{p}\left({ℝ}^{n}\right)$ 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 $\mu$. 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 $\mu$-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