The higher order regularity Dirichlet problem for elliptic systems in the upper-half space

TL;DR

This paper establishes well-posedness of the Dirichlet problem for a broad class of elliptic systems in the upper-half space, using Sobolev space data and nontangential maximal function estimates, including scalar and elasticity systems.

Contribution

It identifies a large class of elliptic systems for which the Dirichlet problem is well-posed with Sobolev space data and maximal function control, extending known results to complex coefficients and elasticity.

Findings

Well-posedness for a broad class of elliptic systems with Sobolev data

Inclusion of scalar and elasticity systems in the class

Use of nontangential maximal operator estimates

Abstract

We identify a large class of constant (complex) coefficient, second order elliptic systems for which the Dirichlet problem in the upper-half space with data in $L^p$-based Sobolev spaces, $1<p<\infty$, of arbitrary smoothness $\ell$, is well-posed in the class of functions whose nontangential maximal operator of their derivatives up to, and including, order $\ell$ is $L^p$-integrable. This class includes all scalar, complex coefficient elliptic operators of second order, as well as the Lam\'e system of elasticity, among others.