Description

• Title:
  Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation $\mathscr{L}[u] = \nabla \mathscr{P}$
• Creator:
  Morrison, George;
  Taheri, Ali
• Description:
  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: $$ \rom{ELS} [(u, \mathscr{P}), \Omega] = \begin{cases} [\nabla u]^t \rom{div} [F_\xi\nabla u ] -F_s [\nabla u]^t u = \nabla \mathscr{P} &\text{in } \Omega, \\ \det \nabla u = 1 &\text{in } \Omega, \\ u \equiv x &\text{on } \partial \Omega. \end{cases} $$% Here $\Omega \subset \R^n$ 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 {\it 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 ({\it 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.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2019 Topological Methods in Nonlinear Analysis
• Date:
  2019.11.09
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 54, No 2 (December 2019); 833 - 862;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.076/23632 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.076/23633
• Subject:
  Nonlinear elliptic systems;
  incompressible twists;
  multiple solutions;
  restricted energies;
  closed geodesics on SO(n);
  self-maps of spheres;
  Hopf degree;
  35J57;
  35J50;
  35J62;
  49J10;
  35A15;
  53C22;
  58D19;
  22E30
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.076
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other