Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the L^p Dirichlet Problem

Abstract

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L₀ = divA⁰(x)∇ + B⁰(x). ∇ is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the L^p Dirichlet problem for the operator L₀ is solvable in the upper half-space ℝ₊ⁿ. In this paper we prove that the L^p solvability is stable under small perturbations of L₀. That is if L₁ is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L₀ and L₁ are sufficiently close in the sense of Carleson measures, then the L^p Dirichlet problem for the operator L₁ is solvable for the same value of p. As a corollary we obtain a new result on L^p solvability of the Dirichlet problem for operators of the form L = divA(x)∇ + B(x) · ∇ where the matrix A satisfies weaker Carleson condition (expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiate and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindoš, Petermichl and Pipher.