$L^p$ regularity of the Dirichlet problem for elliptic equations with singular drift

Abstract

Let L0 and L1 be two elliptic operators in nondivergence form, with coefficients Al and drift terms bl, l = 0, 1 satisfying sup |Y −X|≤ δ(X) 2 |A0 (Y ) − A1 (Y )| 2 + δ (X) 2 |b0 (Y ) − b1 (Y )| 2 δ (X) dX is a Carleson measure in a Lipschitz domain Ω ⊂ Rn+1 , n ≥ 1, (here δ (X) = dist (X, ∂Ω)). If the harmonic measure dωL0 ∈ A∞, then dωL1 ∈ A∞. This is an analog to Theorem 2.17 in [8] for divergence form operators. As an application of this, a new approximation argument and known results we are able to extend the results in [10] for divergence form operators while obtaining totally new results for nondivergence form operators. The theorems are sharp in all cases.