The Lp Dirichlet problem for second-order, non-divergence form operators: solvability and perturbation results

We establish Dahlberg’s perturbation theorem for non-divergence form operators L = A∇2. If L0 and L1 are two operators on a Lipschitz domain such that the Lp Dirichlet problem for the operator L0 is solvable for some p ∈ (1,∞) and the coefficients of the two operators are sufficiently close in the sense of Carleson measure, then the Lp Dirichlet problem for the operator L1 is solvable for the same p. This is a refinement of the A∞ version of this result proved by Rios (2003) in [10]. As a consequence we also improve a result from Dindoš et al. (2007) [4] for the Lp solvability of non-divergence form operators (Theorem 3.2) by substantially weakening the condition required on the coefficients of the operator. The improved condition is exactly the same one as is required for divergence form operators L = divA∇. © 2011 Elsevier Inc. All rights reserved