English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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  the Euclidean n-space (mathbf{X}= lbrace x in {mathbb {R}}^n{:},au003c|x|u003cb 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  symmetries of the critical points and local minimisers of ({mathbb {F}}) over ({mathcal {A}}_phi (mathbf{X}))  the two special cases (n=2) and (n=3). More specifically, that when (n=3), despite the inherent rotational symmetry  the problem, there are NO non-trivial rotationally symmetric critical points of ({mathbb {F}}) over ({mathcal {A}}_phi (mathbf{X})), whereas  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  the planar case (n=2) for Sobolev mappings u  (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,  particular, we show that only  even dimensions do we have an infinitude of non-trivial rotationally symmetric critical points.

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