On the well-posedness of global fully nonlinear first order elliptic systems

In the very recent paper [15], the second author proved that for any f ∈ L2(ℝn,ℝN), the fully nonlinear first order system F(·, Du) = f is well posed in the so-called J. L. Lions space and, moreover, the unique strong solution u: ℝn → ℝN 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 [15] by introducing a new strictly weaker ellipticity condition and by proving well-posedness in the same “energy” space