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An infinite scale of incompressible twisting solutions to the nonlinear elliptic system L [u; A, B] = ?P and the discriminant ?(h, g)

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posted on 2023-06-09, 13:09 authored by George Morrison, Ali TaheriAli Taheri
In this paper we consider the second order nonlinear elliptic system div[A(|x|, |u| 2 , |?u| 2 )?u] + B(|x|, |u| 2 , |?u| 2 )u = [cof ?u]?P, where the unknown vector field u satisfies the incompressibility constraint det ?u = 1 a.e. along with suitable boundary conditions and P = P(x) is an a priori unknown hydrostatic pressure field. Here, A = A(r, s, ?) and B = B(r, s, ?) are sufficiently regular scalar functions satisfying natural structural properties. Most notably in the case of a finite symmetric annulus we prove the existence of a countably infinite scale of topologically distinct twisting solutions to the system in all even dimensions. In sharp contrast in odd dimensions the only twisting solution is the map u = x. We study a related class of systems by introducing the novel notion of a discriminant. Using this a complete and explicit characterisation of all twisting solutions for n = 2 is given and a curious dichotomy in the behaviour of the system and its solutions captured and analyse

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Nonlinear Analysis Theory Methods & Applications

ISSN

0362-546X

Publisher

Elsevier

Volume

173

Page range

209-219

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Analysis and Partial Differential Equations Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-05-04

First Open Access (FOA) Date

2019-05-04

First Compliant Deposit (FCD) Date

2018-05-03

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