Abstract
In this paper we consider a variational problem consisting of an energy functional defined by the integral,
and an associated mapping space, here, the space of incompressible Sobolev mappings of the symmetric annular domain in the Euclidean n-space \(\mathbf{X}= \lbrace x \in {\mathbb {R}}^n{:}\,a<|x|<b \rbrace \):
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.
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Morris, C., Taheri, A. Annular rearrangements, incompressible axi-symmetric whirls and \(L^1\)-local minimisers of the distortion energy. Nonlinear Differ. Equ. Appl. 25, 2 (2018). https://doi.org/10.1007/s00030-017-0485-3
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DOI: https://doi.org/10.1007/s00030-017-0485-3
Keywords
- Annular rearrangements
- \(L^1\)-local minimisers
- Homotopy classes
- Distortion energy
- Symmetric extremisers
- Incompressible whirl mappings