On the uniqueness and monotonicity of energy minimisers in the homotopy classes of incompressible mappings and related problems

Abstract

The goal of this paper is to prove the existence and uniqueness of the so-called energy minimisers in homotopy classes for the variational energy integralF[u;X]=∫XF(|x|2,|u|2)|∇u|2/2dx, with F≥c>0 of class C2 and satisfying suitable conditions and u lying in the Sobolev space of weakly differentiable incompressible mappings of a finite open symmetric plane annulus X onto itself, specifically, lying in A(X)={u∈W1,2(X,R2):det⁡∇u=1a.e.in X, and u≡x on ∂X}. It is well known that the space A(X) admits a countably infinite homotopy class decomposition A(X)=⋃Ak (with k∈Z). We prove that the energy integral F has a unique minimiser in each of these homotopy classes Ak. Furthermore we show that each minimiser is a homeomorphic, monotone, radially symmetric twist mapping of class C3(X‾,X‾) or as smooth as F allows thereafter whilst also being a local minimiser of F over A(X) with respect to the L1-metric. To our best knowledge this is the first uniqueness result for minimisers in homotopy classes in the context of incompressible mappings.