The Dirichlet problem for elliptic equations in divergence and nondivergence form with singular drift term
Abstract
Given two elliptic operators L and M in nondivergence form, with coefficients AL(x), AM(x) and drift terms bL(x), bM(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in Rn+1, then their harmonic measures are mutually absolutely continuous. As an application of this, a new approximation argument and known results we obtain necessary and sufficient conditions for a single operator L (in divergence or nondivergence form) to have regular harmonic measure with respect to Lebesgue measure. The results are sharp in all cases.
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