Boundary value problems for second order elliptic operators with complex coefficients

Martin Dindos, Jill Pipher

Research output: Contribution to journalArticlepeer-review

Abstract

The theory of second order complex coeffcient operators of the
form L = divA(x)r has recently been developed under the assumption of
p-ellipticity. In particular, if the matrix A is p-elliptic, the solutions u to
Lu = 0 will satisfy a higher integrability, even though they may not be
continuous in the interior. Moreover, these solutions have the property that
jujp=2-1u 2 W1;2
loc . These properties of solutions were used by Dindos-Pipher
to solve the Lp Dirichlet problem for p-elliptic operators whose coeffcients
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.
Original languageEnglish
Pages (from-to)1897–1938
Number of pages41
JournalAnalysis and PDE
Volume13
Issue number6
DOIs
Publication statusPublished - 12 Sep 2020

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