On $L^2$ Solvability of BVPs for elliptic systems

TL;DR

This paper establishes $L^2$ solvability for boundary value problems of certain elliptic systems in the upper half-space, utilizing a first order formalism and proving partial Rellich inequalities.

Contribution

It advances the understanding of $L^2$ boundary value problems for elliptic systems using a novel approach that proves partial Rellich inequalities without full boundary estimates.

Findings

Proved $L^2$ solvability for elliptic systems with transversally independent coefficients.

Applied the first order formalism of Auscher-Axelsson-McIntosh.

Established partial Rellich boundary inequality results.

Abstract

In this article we prove solvability results for $L^2$ boundary value problems of some elliptic systems $Lu=0$ on the upper half-space $\R^{n+1}_{+}, n\ge 1$, with transversally independent coefficients. We use the first order formalism introduced by Auscher-Axelsson-McIntosh and further developed with a better understanding of the classes of solutions in the subsequent work of Auscher-Axelsson. The interesting fact is that we prove only half of the Rellich boundary inequality without knowing the other half.