The Lp Dirichlet and regularity problems for second order elliptic systems with application to the Lame system

In a recent joint paper with Hwang and Mitrea we have introduced new solvability methods for strongly elliptic second order systems in divergence form 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 it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the L-p Dirichlet problem to the interval 2 - epsilon < p < 2(n-1)/(n-3) + epsilon, for systems in dimension n = 2, 3 in the range 2 - epsilon < p < infinity. We do this by considering solvability of the Regularity problem (with boundary data having one derivative in L-p) in the range 2 - epsilon < p < 2 + epsilon. Secondly, we look at perturbation type-results where we can deduce solvability of the L-p Dirichlet problem for one operator from known L-p Dirichlet solvability of a "close" operator (in the sense of Carleson measure). This leads to improvement of the main result of our previous paper; we establish solvability of the L-p Dirichlet prob lem in the interval 2 - epsilon < p < 2(n-1)/(n-2) + epsilon under a much weaker (oscillation-type) Carleson condition. A particular example of the system where all these results apply is the Lame operator for isotropic inhomogeneous materials with Poisson ratio nu < 0.396. In this specific case further improvements of the solvability range are possible.