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

TL;DR

This paper proves that the solvability of the L^p Dirichlet problem for certain second-order non-divergence form operators is stable under small perturbations, improving previous results by weakening coefficient conditions.

Contribution

It extends Dahlberg's perturbation theorem to non-divergence form operators with weaker coefficient conditions, matching those for divergence form operators.

Findings

Established L^p solvability stability under Carleson measure perturbations.

Improved coefficient conditions for L^p solvability of non-divergence form operators.

Extended previous perturbation results to a broader class of operators.

Abstract

We establish Dahlberg's perturbation theorem for non-divergence form operators L = A\nabla^2. If L_0 and L_1 are two operators on a Lipschitz domain such that the L^p Dirichlet problem for the operator L_0 is solvable for some p in (1,\infty) and the coefficients of the two operators are sufficiently close in the sense of Carleson measure, then the L^p Dirichlet problem for the operator L_1 is solvable for the same p. This is an improvement of the A_{\infty} version of this result proved by Rios in "The L^p Diriclet problem and nondivergence harmonic measure" (Trans. AMS 355, 2 (2003)). As a consequence we also improve a result from Dindos, Petermichl and Pipher, "The L^p Dirichlet problem for second order elliptic operators and a p-adapted square function" (J. Fun. Anal. 249 (2007)) for the L^p solvability of non-divergence form operators 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 = div A\nabla.