Twist maps as energy minimisers in homotopy classes: Symmetrisation and the coarea formula. (March 2017)

• Record Type: Journal Article
• Authors: Morris, C.; Taheri, A.
• Is Part Of: Nonlinear analysis. Volume 152(2017)
• Page Start: 250
• Page End: 275
• Publication Date: 2017-03
• Subjects: Symmetrisation -- Coarea formula -- Minimisers in homotopy classes -- L1-local minimisers -- Twist maps -- Incompressible Sobolev maps; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.12.014
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let X = X [ a, b ] = { x : a < | x | < b } ⊂ R n with 0 < a < b < ∞ fixed be an open annulus and consider the energy functional, F [ u ; X ] = 1 2 ∫ X | ∇ u | 2 | u | 2 d x, over the space of admissible incompressible Sobolev maps A ϕ ( X ) = { u ∈ W 1, 2 ( X, R n ) : det ∇ u = 1 a.e. in X and u | ∂ X = ϕ }, where ϕ is the identity map of X ¯ . Motivated by the earlier works (Taheri (2005), (2009)) in this paper we examine the twist maps as extremisers of F over A ϕ ( X ) and investigate their minimality properties by invoking the coarea formula and a symmetrisation argument. In the case n = 2 where A ϕ ( X ) is a union of infinitely many disjoint homotopy classes we establish the minimality of these extremising twists in their respective homotopy classes a result that then leads to the latter twists being L 1 -local minimisers of F in A ϕ ( X ) . We discuss variants and extensions to higher dimensions as well as to related energy functionals.