Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

We prove that the Dirichlet problem for degenerate elliptic equations $\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }u)=0$ in the upper half-space $(x,t)\in $\mathbb{R}$^{n+1}+$issolvablewhen$n\geq2$andtheboundarydataisin$L^p\mu($\mathbb{R}$^n)$forsome$p<\infty$\mathrm{.}Thecoefficientmatrix$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 $A_{\mathrm{\infty }}$ with respect to the $\mu$-weighted Lebesgue measure on ${\mathbb{R}}^n$. The Carleson measure estimate allows us to avoid applying the method of $ϵ$-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.