Boundary value problems for second-order elliptic operators with complex coefficients

Dindoš, Martin; Pipher, Jill

doi:10.2140/apde.2020.13.1897

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Boundary value problems for second-order elliptic operators with complex coefficients

Martin Dindoš and Jill Pipher

Vol. 13 (2020), No. 6, 1897–1938

DOI: 10.2140/apde.2020.13.1897

Abstract

The theory of second-order complex-coefficient operators of the form $ℒ = div A (x) \nabla$ has recently been developed under the assumption of $p$ -ellipticity. In particular, if the matrix $A$ is $p$ -elliptic, the solutions $u$ to $ℒ u = 0$ will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that $| u |^{p ∕ 2 - 1} u \in W_{loc}^{1, 2}$ . These properties of solutions were used by Dindoš and Pipher to solve the $L^{p}$ Dirichlet problem for $p$ -elliptic operators whose coefficients satisfy a further regularity condition, a Carleson measure condition that has often appeared in the literature in the study of real, elliptic divergence form operators. This paper contains two main results. First, we establish solvability of the regularity boundary value problem for this class of operators, in the same range as that of the Dirichlet problem. The regularity problem, even in the real elliptic setting, is more delicate than the Dirichlet problem because it requires estimates on derivatives of solutions. Second, the regularity results allow us to extend the previously established range of $L^{p}$ solvability of the Dirichlet problem using a theorem due to Z. Shen for general bounded sublinear operators.

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Keywords

complex-coefficient elliptic PDEs, boundary value problems, $p$-ellipticity

Mathematical Subject Classification 2010

Primary: 35J25

Milestones

Received: 21 October 2018

Revised: 23 June 2019

Accepted: 13 August 2019

Published: 12 September 2020

Authors

Martin Dindoš
School of Mathematics The University of Edinburgh and Maxwell Institute of Mathematical Sciences Edinburgh United Kingdom

Jill Pipher
Department of Mathematics Brown University Providence, RI United States

Vol. 13, No. 6, 2020