An infinite scale of incompressible twisting solutions to the nonlinear elliptic system [formula omitted] and the discriminant [formula omitted

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=1a.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.