On critical points of the [formula omitted]-energy over a space of measure preserving maps

Abstract

Let X⊂Rn be a bounded Lipschitz domain and consider the σ2-energy functional Fσ2[u;X]:=∫X|∇u∧∇u|2dx, over the space of admissible maps A(X)={u∈W1,4(X,Rn):u|∂X=x, det∇u=1 for Ln-a.e. in X}. A good measure of how much a map u stretches areas (of 2-dimensional sub-manifolds of the domain X) is the norm of ∇u∧∇u:∧2TX→∧2TX, analogously to |∇u|2 (the Dirichlet energy density) that is a measure of length’s stretching. These kinds of functionals also were arisen as a physical model describing the strong interactions of quantum fields which was introduced by T. Skyrme in 1961. 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 Fσ2[⋅;X] over A(X). In particular we present a novel characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions which follows very closely to the ideas that have been introduced by the second author in his recent paper Shahrokhi-Dehkordi and Taheri (2009) [17]. Indeed we show that in even dimensions the latter system of equations admits 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. We investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula.