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Generalised twists, stationary loops and the Dirichlet energy over a space of measure preserving maps
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posted on 2023-06-08, 05:34 authored by M S Shahrokhi-Dehkordi, Ali TaheriAli TaheriLet $${\Omega \subset \mathbb{R}^n}$$ be a bounded Lipschitz domain and consider the Dirichlet energy functional $${\mathbb F} [{\bf u}, \Omega] := \frac{1}{2} \int\limits_\Omega|
abla {\bf u}({\bf x})|^2 \, d{\bf x},$$ over the space of measure preserving maps $${\mathcal A}(\Omega)=\left\{{\bf u}\in W^{1,2}(\Omega, \mathbb{R}^n) : {\bf u}|_{\partial \Omega} = {\bf x}, \mbox{ }\det
abla {\bf u} = 1 \mbox{ }{{\rm a.e}.\; {\rm in} \Omega}\right\}.$$ In this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler¿Lagrange equations associated with $${{\mathbb F}}$$ over $${{\mathcal A}(\Omega)}$$ . The main result here is that in even dimensions the latter equations admit infinitely many solutions, modulo isometries, amongst such maps. We investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula.
abla {\bf u}({\bf x})|^2 \, d{\bf x},$$ over the space of measure preserving maps $${\mathcal A}(\Omega)=\left\{{\bf u}\in W^{1,2}(\Omega, \mathbb{R}^n) : {\bf u}|_{\partial \Omega} = {\bf x}, \mbox{ }\det
abla {\bf u} = 1 \mbox{ }{{\rm a.e}.\; {\rm in} \Omega}\right\}.$$ In this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler¿Lagrange equations associated with $${{\mathbb F}}$$ over $${{\mathcal A}(\Omega)}$$ . The main result here is that in even dimensions the latter equations admit infinitely many solutions, modulo isometries, amongst such maps. We investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula.
History
Publication status
- Published
Journal
Calculus of Variations and Partial Differential EquationsISSN
0944-2669Publisher
Springer VerlagExternal DOI
Issue
2Volume
35Page range
191-213Pages
13.0Department affiliated with
- Mathematics Publications
Notes
SOME OF EQUATION LOST IN ABSTRACTFull text available
- No
Peer reviewed?
- Yes
Legacy Posted Date
2012-02-06Usage metrics
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