Perturbation theory for solutions to second order elliptic operators with complex coefficients and the $L^p$ Dirichlet problem

TL;DR

This paper proves that the solvability of the $L^p$ Dirichlet problem for second order elliptic operators with complex coefficients remains stable under small perturbations, extending previous results to weaker coefficient conditions.

Contribution

It establishes the stability of $L^p$ solvability under small Carleson measure perturbations for complex coefficient elliptic operators, with weaker regularity assumptions on the coefficients.

Findings

$L^p$ solvability is stable under small Carleson measure perturbations.

Weaker Carleson conditions suffice for $L^p$ solvability.

Extension of real case results to complex coefficient operators.

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 ${\mathcal L}_0=\mbox{div} A^0(x)\nabla+B^0(x)\cdot\nabla$ is a $p$-elliptic operator satisfying certain Carleson condition on $\nabla A$ and $B$ then the $L^p$ Dirichlet problem for the operator ${\mathcal L}_0$ is solvable in the upper half-space ${\mathbb R}^n_+$. In this paper we prove that the $L^p$ solvability is stable under small perturbations of ${\mathcal L}_0$. That is if ${\mathcal L}_1$ is another divergence form elliptic operator with complex coefficients and the coefficients of the operators ${\mathcal L}_0$ and ${\mathcal L}_1$ are sufficiently close in the sense of Carleson measures (considering the differences of coefficients), then the $L^p$ Dirichlet problem for the operator ${\mathcal L}_1$ 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 ${\mathcal L}=\mbox{div} A(x)\nabla+B(x)\cdot\nabla$ where the matrix $A$ satisfies weaker Carleson condition than in our earlier paper; in particular the coefficients of $A$ need no longer be differentiable 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 Dindo\v{s}, Petermichl and Pipher.