The $L^p$ Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lam\'e system

Abstract

In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains 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 $2-\varepsilon < p<\frac{2(n-1)}{(n-3)}+\varepsilon$, for systems in dimension $n=2,3$ in the range $2-\varepsilon < p<\infty$. We do this by considering solvability of the Regularity problem (with boundary data having one derivative in $L^p$) in the range $2-\varepsilon < p<2+\varepsilon$. 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 \lq\lq close" operator (in the sense of Carleson measure). This leads to improvement of the main result of the paper arXiv:1708.02289; we establish solvability of the $L^p$ Dirichlet problem in the interval $2-\varepsilon < p<\frac{2(n-1)}{(n-2)}+\varepsilon$ under a much weaker (oscillation-type) Carleson condition. A particular example of the system where all these results apply is the Lam\'e operator for isotropic inhomogeneous materials with Poisson ratio $\nu<0.396$. In this specific case further improvements of the solvability range are possible, see the upcoming work with J. Li and J. Pipher.