Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation L [u] = ?P

In this paper we address questions on the existence and multiplicity of a class of geometrically motivated mappings with certain symmetries that serve as solutions to the nonlinear system in variation: [equation not shown] Here ? ? R n is a bounded domain, F = F(r, s, ?) is a sufficiently smooth Lagrangian, Fs = Fs(|x|, |u| 2 , |?u| 2 ) and F? = F?(|x|, |u| 2 , |?u| 2 ) with Fs and F? denoting the derivatives of F with respect to the second and third variables respectively while P is an a priori unknown hydrostatic pressure resulting from the incompressibility constraint det ?u = 1. Among other things, by considering twist mappings u with an SO(n)-valued twist path, we prove the existence of multiple and topologically distinct solutions to ELS for n = 2 even versus the only (non) twisting solution u = x for n = 3 odd. An extremality analysis for twist paths and those of Lie exponential types and a suitable formulation of a differential operator action on twists relating to ELS are the key ingredients in the proof.