The $L^p$ Dirichlet boundary problem for second order Elliptic Systems with rough coefficients

TL;DR

This paper proves the solvability of the $L^p$ Dirichlet problem for second-order elliptic systems with rough coefficients on Lipschitz domains, extending previous scalar results to systems like the Lamé system in inhomogeneous materials.

Contribution

It introduces solvability results for elliptic systems with Carleson measure coefficients, applicable to systems like the Lamé system, with new estimates for solutions in rough coefficient settings.

Findings

Solvability for $L^p$ boundary data near $p=2$ on Lipschitz domains.

Extension of scalar Carleson condition results to elliptic systems.

Application to isotropic inhomogeneous materials with specific Poisson ratios.

Abstract

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in an interval of the form $\big(2-\varepsilon,\frac{2(n-1)}{n-2}+\varepsilon\big)$ for some small $\varepsilon>0$). 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 significant example of a system to which our result may be applied is the Lam\'e system for isotropic inhomogeneous materials. We show that our result applies to isotropic materials with Poisson ratio $\nu<0.396$. Dealing with genuine systems gives rise to substantial new challenges, absent in the scalar case. Among other things, there is no maximum principle for general elliptic systems, and the De Giorgi - Nash - Moser theory may also not apply. We are, nonetheless, successful in establishing estimates for the square-function and the nontangential maximal operator for the solutions of the elliptic system described earlier, and use these as alternative tools for proving $L^p$ solvability results for $p$ near $2$.