On a class of stationary loops on SO ( n ) $\mathbf{SO}(n)$ and the existence of multiple twisting solutions to a nonlinear elliptic system subject to a hard incompressibility constraint

• Published in: Boundary value problems Vol. 2018; no. 1; pp. 1 - 24
• Main Authors: Morrison, George, Taheri, Ali
• Format: Journal Article
• Language: English
• Published: SpringerOpen 22.08.2018
• Subjects: Curl-free vector fields; Geodesics on SO ( n ) $\mathbf{SO}(n); Incompressible twists; Multiple stationary loops; Nonlinear elliptic systems; Weighted Dirichlet type Lagrangians
• ISSN: 1687-2770; 1687-2770
• DOI: 10.1186/s13661-018-1047-2

Abstract In this paper we consider the second order nonlinear elliptic system in divergence and variational form {div[Fξ(|x|,|∇u|2)∇u]=[cof∇u]∇Pin U,det∇u=1in U,u=φon ∂U, $$\begin{aligned} \textstyle\begin{cases} \operatorname{div}[ F_{\xi }(\vert x\vert ,\vert \nabla u\vert ^{2})\nabla u ] = [ \operatorname{cof}\nabla u] \nabla \mathscr{P} &\text{in }U, \\ \operatorname{det}\nabla u = 1 &\text{in }U, \\ u = \varphi &\text{on }\partial U, \end{cases}\displaystyle \end{aligned}$$ where F=F(r,ξ) $F=F(r,\xi)$ is a sufficiently regular Lagrangian satisfying suitable structural properties and P $\mathscr{P}$ is an a priori unknown Lagrange multiplier. Most notably, for a finite symmetric n-annulus, we prove the existence of an infinite family of monotone twisting solutions to this system in all even dimensions by linking the system to a set of nonlinear isotropic ODEs on the Lie group SO(n) $\mathbf{SO}(n)$. We prove the existence of multiple closed stationary loops in the geodesic form Q(r)=exp{f(r)H} $\mathbf{Q}(r) = \operatorname{exp}\{f(r) \mathbf{H}\}$ with H∈so(n) $\mathbf{H}\in \mathfrak{so}(n)$ to these ODEs that remarkably serve as the twist loops associated with the desired twisting solutions u to the above system. An analysis of curl-free vector fields generated by symmetric matrix fields plays a pivotal role.