Polyconvexity, generalised twists and energy minimizers on a space of self-maps of annuli in the multi-dimensional calculus of variations

M. S. Shahrokhi-Dehkordi; A. Taheri

doi:10.1515/ACV.2009.014

Published by De Gruyter July 22, 2009

Polyconvexity, generalised twists and energy minimizers on a space of self-maps of annuli in the multi-dimensional calculus of variations

M. S. Shahrokhi-Dehkordi and A. Taheri

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/ACV.2009.014

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Abstract

Let be a bounded Lipschitz domain and consider the energy functional

over the space of admissible maps

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 𝔽 over . The main result is a complete characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions. Indeed we show that in even dimensions the latter system of equations admit infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler–Lagrange equations.