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We establish solvability methods for strongly elliptic second order systems in divergence form with lower order (drift) terms on a domain above a Lipschitz graph, satisfying L p -boundary data for p near 2. The main novel aspect of our result is that the coefficients of the operator do not have to be constant or have very high regularity, instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A particular example of a system where this result can be applied is the Lamé operator for isotropic inhomogeneous materials.The systems case poses substantial new challenges not present in the scalar case. In particular, there is no maximum principle for general elliptic systems and De Giorgi -Nash -Moser theory is also not available. Despite this we are able to establish estimates for the square and nontangential maximal functions for the solution of the elliptic system and use these estimates to establish L p solvability for p near 2.

We establish a new theory of regularity for elliptic complex valued second order equations of the form L =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 Hölder 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 Dragičević 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 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 L =divA(∇·)+B·∇ when A and B satisfy a Carleson measure condition, which previously was known only in the real valued case. We show solvability of the L 2 Dirichlet problem, as well as solvability of the L p Dirichlet boundary value problem for p in the range where A is p-elliptic.

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