Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

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.