Annular rearrangements, incompressible axi-symmetric whirls and \(L^1\)-local minimisers of the distortion energy

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Abstract

In this paper we consider a variational problem consisting of an energy functional defined by the integral,
$$\begin{aligned} \mathbb {F}[u,\mathbf{X}] = \frac{1}{2}\int _{\mathbf{X}} \frac{|\nabla u|^2}{|u|^2} \,dx, \end{aligned}$$
and an associated mapping space, here, the space of incompressible Sobolev mappings of the symmetric annular domain in the Euclidean n-space \(\mathbf{X}= \lbrace x \in {\mathbb {R}}^n{:}\,a<|x|<b \rbrace \):
$$\begin{aligned} \mathcal {A}_{\phi }(\mathbf{X}) = \bigg \lbrace u \in W^{1,2}(\mathbf{X},{\mathbb {R}}^n){:}\,\det \nabla u = 1 \,\, a.e. \text { and } u |_{_{\partial \mathbf{X}}} \equiv x \bigg \rbrace . \end{aligned}$$
The goal is then twofold. Firstly to establish and highlight an unexpected difference in symmetries of the critical points and local minimisers of \({\mathbb {F}}\) over \({\mathcal {A}}_\phi (\mathbf{X})\) in the two special cases \(n=2\) and \(n=3\). More specifically, that when \(n=3\), despite the inherent rotational symmetry in the problem, there are NO non-trivial rotationally symmetric critical points of \({\mathbb {F}}\) over \({\mathcal {A}}_\phi (\mathbf{X})\), whereas in sharp contrast, when \(n=2\), not only that there is an infinitude of rotationally symmetric critical points of the energy but also there is an infinitude of local minimisers of \({\mathbb {F}}\) over \({\mathcal {A}}_\phi (\mathbf{X})\) with respect to the \(L^1\)-metric. At the heart of this analysis is an investigation into the rich homotopy structure of the space of self-mappings of annuli. The second aim is to introduce and implement a novel symmetrisation technique in the planar case \(n=2\) for Sobolev mappings u in \(\mathcal {A}_\phi (\mathbf{X})\) that lowers the energy whilst keeping the homotopy class of u invariant. We finally generalise and extend some of these results to higher dimensions, in particular, we show that only in even dimensions do we have an infinitude of non-trivial rotationally symmetric critical points.

Keywords

Annular rearrangements \(L^1\)-local minimisers Homotopy classes Distortion energy Symmetric extremisers Incompressible whirl mappings 

Mathematics Subject Classification

35J57 35Q74 49J19 49K20 49Q20 22E30 58C35 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SussexFalmer, BrightonEngland, UK

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