Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations

Abstract

We prove that the Dirichlet problem for degenerate elliptic equations div(A∇u) = 0 in the upper half-space (x, t) ∈ R n+1 + is solvable when n ≥ 2 and the boundary data is in L p µ (R n ) 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 A 2 -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 R n . The Carleson measure estimate allows us to avoid applying the method of ǫ-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.

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