Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

We prove that the Dirichlet problem for degenerate elliptic equations div(A∇u)=0 in the upper half-space (x,t)∈Rⁿ⁺¹₊ is solvable when n≥2 and the boundary data is in L^p_μ(Rⁿ) 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 A₂-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 Rⁿ. 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.