An infinite scale of incompressible twisting solutions to the nonlinear elliptic system ℒ[u;A, B]=∇P and the discriminant Δ(h, g). (August 2018)

• Record Type: Journal Article
• Authors: Morrison, George; Taheri, Ali
• Is Part Of: Nonlinear analysis. Volume 173(2018)
• Page Start: 209
• Page End: 219
• Publication Date: 2018-08
• Subjects: Multiple solutions -- Nonlinear elliptic systems -- Incompressible twists -- Geodesics on SO(n) -- Symmetries and Discriminants; 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.2018.04.002
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we consider the second order nonlinear elliptic system div [ A ( | x |, | u | 2, | ∇ u | 2 ) ∇ u ] + B ( | x |, | u | 2, | ∇ u | 2 ) u = [ cof ∇ u ] ∇ P, where the unknown vector field u satisfies the incompressibility constraint det ∇ u = 1 a.e. along with suitable boundary conditions and P = P ( x ) is an a priori unknown hydrostatic pressure field. Here, A = A ( r, s, ξ ) and B = B ( r, s, ξ ) are sufficiently regular scalar functions satisfying natural structural properties. Most notably in the case of a finite symmetric annulus we prove the existence of a countably infinite scale of topologically distinct twisting solutions to the system in all even dimensions. In sharp contrast in odd dimensions the only twisting solution is the map u ≡ x . We study a related class of systems by introducing the novel notion of a discriminant . Using this a complete and explicit characterisation of all twisting solutions for n ≥ 2 is given and a curious dichotomy in the behaviour of the system and its solutions captured and analysed.