Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

TL;DR

This paper establishes solvability of the Dirichlet problem for certain degenerate elliptic equations in the upper half-space with boundary data in weighted Lebesgue spaces, using Carleson measure estimates to handle degeneracy and non-symmetry.

Contribution

It introduces a novel approach using Carleson measure estimates to prove solvability for degenerate elliptic equations with minimal coefficient regularity, extending previous methods.

Findings

Solvability of Dirichlet problem for degenerate elliptic equations with $A_2$-weight control.

Carleson measure estimates imply the degenerate elliptic measure is in $A_$.

Method simplifies previous proofs by avoiding $$-approximability techniques.

Abstract

We prove that the Dirichlet problem for degenerate elliptic equations $\mathrm{div}(A \nabla u) = 0$ in the upper half-space $(x,t)\in \mathbb{R}^{n+1}_+$ is solvable when $n\geq2$ and the boundary data is in $L^p_\mu(\mathbb{R}^n)$ 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 $\mu$. 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 $\mathbb{R}^n$. The Carleson measure estimate allows us to avoid applying the method of $\epsilon$-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.