Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph