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

In the very recent paper [K1], the second author proved that for any $ f\in L^2($\mathbb{R}$^n,$\mathbb{R}$^N)$,thefullynonlinearfirstordersystem$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.