Vol. 13, No. 6, 2020

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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
Abstract

The theory of second-order complex-coefficient operators of the form = divA(x) has recently been developed under the assumption of p-ellipticity. In particular, if the matrix A is p-elliptic, the solutions u to 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|p21u Wloc1,2. These properties of solutions were used by Dindoš and Pipher to solve the Lp 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 Lp solvability of the Dirichlet problem using a theorem due to Z. Shen for general bounded sublinear operators.

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