Existence and uniqueness of global solutions to fully nonlinear first order elliptic systems

Let F:Rn×RN×n→RNF:Rn×RN×n→RN be a Carathéodory map. In this paper we consider the problem of existence and uniqueness of weakly differentiable global strong a.e. solutions u:Rn⟶RNu:Rn⟶RN to the fully nonlinear PDE system equation(1)F(⋅,Du)=f,a.e. on  Rn, when f∈L2(Rn)Nf∈L2(Rn)N. By introducing an appropriate notion of ellipticity, we prove the existence of solution to (1) in a tailored Sobolev “energy” space (known also as the J.L. Lions space) and a uniqueness a priori estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods and a “perturbation device” which allows the use of Campanato’s notion of near operators, an idea developed for the 2nd order case.