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

Martin Dindos, Jill Pipher

Research output: Contribution to journalArticlepeer-review

Abstract

We establish a new theory of regularity for elliptic complex valued second order equations of the form =divA(∇⋅), when the coefficients of the matrix A satisfy a natural algebraic condition, a strengthened version of a condition known in the literature as L^p-dissipativity. Precisely, the regularity result is a reverse Holder condition for Lp averages of solutions on interior balls, and serves as a replacement for the De Giorgi - Nash - Moser regularity of solutions to real-valued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studied necessary and sufficient conditions for L^p-dissipativity of second order complex coefficient operators and systems. Recently, Carbonaro and Dragi\v{c}evi\'c introduced a condition they termed p-ellipticity, and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators. Their p-ellipticity condition is exactly our strengthened version of Lp-dissipativity. The regularity results of the present paper are applied to solve Lp Dirichlet problems for =divA(∇⋅)+B⋅∇ when A and B satisfy a natural and familiar Carleson measure condition. We show solvability of the Lp Dirichlet boundary value problem for p in the range where A is p-elliptic.
Original languageEnglish
Pages (from-to)255-298
Number of pages38
JournalAdvances in Mathematics
Volume341
Early online date30 Oct 2018
DOIs
Publication statusPublished - 7 Jan 2019

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