Existence and Uniqueness of Global Solutions to Fully Nonlinear First Order Elliptic Systems

Let $F:{\mathbb{R}}^n\times {\mathbb{R}}^{N\times n}\to {\mathbb{R}}^N$ be a Caratheodory map. In this paper we consider the problem of existence and uniqueness of weakly differentiable global strong a.e. solutions $u: $\mathbb{R}$^n \longrightarrow $\mathbb{R}$^N$ to the fully nonlinear PDE system $$\begin{array}{cccc}& \text{label}1F(\cdot ,Du)=f,\text{ a.e. on }{\mathbb{R}}^n,& & \text{(1)}\end{array}$$ when $f\in L^2({\mathbb{R}}^n{)}^N$. This problem has not been considered before. By introducing an appropriate notion of ellipticity, we prove existence of solution to \eqref{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 to use of Campanato’s notion of near operators, an idea developed for the 2nd order case.