Existence and uniqueness of global solutions to fully nonlinear first order elliptic systems. (March 2015) - Katzourakis, Nikos
Abstract: Let F : R n × R N × n → R N be a Carathéodory map. In this paper we consider the problem of existence and uniqueness of weakly differentiable global strong a.e. solutions u : R n ⟶ R N to the fully nonlinear PDE system (1) F ( ⋅, D u ) = f, a.e. on  R n, when f ∈ L 2 ( R n ) 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.