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

Let $F : \mathbb{R}^n \times \mathbb{R}^{N\times n} \rightarrow \mathbb{R}^N$ 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 \[\label{1} \tag{1} F(\cdot,Du ) \,=\, f, \ \ \text{ a.e. on }\mathbb{R}^n, \] 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.