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

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 $u:{\mathbb{R}}^n⟶{\mathbb{R}}^N$ 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.

arXiv commentJournal: Advances in Nonlinear Analysis, 13 pages