We prove that the Dirichlet problem for degenerate elliptic equations in the upper half-space is solvable when and the boundary data is in for some . The coefficient matrix is only assumed to be measurable, real-valued and -independent with a degenerate bound and ellipticity controlled by an -weight . 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 with respect to the -weighted Lebesgue measure on . The Carleson measure estimate allows us to avoid applying the method of -approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.
"Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations." Anal. PDE 12 (8) 2095 - 2146, 2019. https://doi.org/10.2140/apde.2019.12.2095