On the Well-Posedness of Global Fully Nonlinear First Order Elliptic Systems

TL;DR

This paper extends previous work on fully nonlinear first order elliptic systems by introducing a weaker ellipticity condition and proving well-posedness in the same energy space.

Contribution

It introduces a new, weaker ellipticity condition for nonlinear first order systems and establishes their well-posedness in the J.L. Lions space.

Findings

Established well-posedness under the new ellipticity condition

Extended previous results to a broader class of systems

Provided quantitative estimates for solutions

Abstract

In the very recent paper [K1], the second author proved that for any $ f\in L^2(\mathbb{R}^n,\mathbb{R}^N)$, the fully nonlinear first order system $F(\cdot,\mathrm{D} u) =f$ is well posed in the so-called J.L. Lions space and moreover the unique strong solution $u:\mathbb{R}^n\longrightarrow \mathbb{R}^N$ to the problem satisfies a quantitative estimate. A central ingredient in the proof was the introduction of an appropriate notion of ellipticity for $F$ inspired by Campanato's classical work in the 2nd order case. Herein we extend the results of [K1] by introducing a new strictly weaker ellipticity condition and by proving well posedness in the same "energy" space.