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^pdissipativity. 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 realvalued divergence form elliptic operators. In a series of papers, Cialdea and Maz'ya studied necessary and sufficient conditions for L^pdissipativity of second order complex coefficient operators and systems. Recently, Carbonaro and Dragi\v{c}evi\'c introduced a condition they termed pellipticity, and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators. Their pellipticity condition is exactly our strengthened version of Lpdissipativity. 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 pelliptic.
Original language  English 

Pages (fromto)  255298 
Number of pages  38 
Journal  Advances in Mathematics 
Volume  341 
Early online date  30 Oct 2018 
DOIs  
Publication status  Published  7 Jan 2019 
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Martin Dindos
 School of Mathematics  Personal Chair of harmonic analysis and partial differential
Person: Academic: Research Active