Existence and uniqueness of global solutions to fully nonlinear first order elliptic systems. (March 2015)

• Record Type: Journal Article
• Authors: Katzourakis, Nikos
• Is Part Of: Nonlinear analysis. Volume 115(2015)
• Page Start: 50
• Page End: 61
• Publication Date: 2015-03
• Subjects: primary 35J46 35J47 35J60 -- secondary 35D30 32A50 32W50; Cauchy–Riemann equations -- Fully nonlinear systems -- Elliptic first order systems -- Calculus of variations -- Campanato's near operators -- Cordes' condition -- Compensated compactness -- Baire category method -- Convex Integration; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.12.002
• Languages: English
• ISSNs: 0362-546X

Abstract

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.