Perturbation theory for solutions to second order elliptic operators with complex coefficients and the Lp Dirichlet problem

Martin Dindos, Jill Pipher

Research output: Contribution to journalArticlepeer-review

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 Lp Dirichlet problem for the operator L0 is solvable in the upper half-space Rn+. In this paper we prove that the Lp 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 Lp Dirichlet problem for the operator L1 is solvable for the same value of p. As a corollary we obtain a new result on Lp 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 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 Dindos, Petermichl and Pipher.
Original languageEnglish
Pages (from-to)749-770
Number of pages22
JournalActa Mathematica Sinica
Volume35
Issue number6
Early online date20 May 2019
DOIs
Publication statusPublished - 30 Jun 2019

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