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

Morrison, George and Taheri, Ali (2019) Topology of twists, extremising twist paths and multiple solutions to the nonlinear system in variation L [u] = ∇P. Topological Methods in Nonlinear Analysis. ISSN ISSN 1230-3429 (Accepted)

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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:
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Here Ω ⊂ R n is a bounded domain, F = F(r, s, ξ) is a sufficiently smooth Lagrangian, Fs = Fs(|x|, |u| 2 , |∇u| 2 ) and Fξ = Fξ(|x|, |u| 2 , |∇u| 2 ) with Fs and Fξ 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Related URLs:
Depositing User: Ali Taheri
Date Deposited: 26 Apr 2019 13:37
Last Modified: 01 Jul 2019 16:00
URI: http://sro.sussex.ac.uk/id/eprint/83413

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