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

Abstract: We establish a new theory of regularity for elliptic complex valued second order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, 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 H\"older condition for $L^p$ 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 $L^p$-dissipativity. The regularity results of the present paper are applied to solve $L^p$ Dirichlet problems for $\mathcal L=$div$A(\nabla\cdot)+B\cdot\nabla$ when $A$ and $B$ satisfy a natural and familiar Carleson measure condition. We show solvability of the $L^p$ Dirichlet boundary value problem for $p$ in the range where $A$ is $p$-elliptic.