Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation $\mathscr{L}[u] = \nabla \mathscr{P}$

Abstract

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: $$ {\rm ELS} [(u, \mathscr{P}), \Omega] = \begin{cases} [\nabla u]^t {\rm div} [F_\xi\nabla u ] -F_s [\nabla u]^t u = \nabla \mathscr{P} &\text{in } \Omega, \\ \det \nabla u = 1 &\text{in } \Omega, \\ u \equiv x &\text{on } \partial \Omega. \end{cases} $$ Here $\Omega \subset \mathbb R^n$ is a bounded domain, $F=F(r, s, \xi)$ is a sufficiently smooth Lagrangian, $F_s=F_s(|x|, |u|^2, |\nabla u|^2)$ and $F_\xi=F_\xi(|x|, |u|^2, |\nabla u|^2)$ with $F_s$ and $F_\xi$ denoting the derivatives of $F$ with respect to the second and third variables respectively while $\mathscr{P}$ is an a priori unknown hydrostatic pressure resulting from the incompressibility constraint $\det \nabla u =1$. 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 $n \ge 2$ even versus the only (non) twisting solution $u \equiv x$ for $n \ge 3$ 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.