Stationary twists and energy minimizers on a space of measure preserving maps Original Research Article

Let Ω⊂RnΩ⊂Rn be a bounded Lipschitz domain, View the MathML sourceF:Rn×n→R a suitably quasiconvex integrand and consider the energy functional View the MathML sourceF[u,Ω]≔∫ΩF(∇u), Turn MathJax on over the space of measure preserving maps View the MathML sourceAp(Ω)={u∈W1,p(Ω,Rn):u|∂Ω=x,det∇u=1 a.e. in Ω}. Turn MathJax on In this paper we discuss the question of existence of multiple strong local minimizers for View the MathML sourceF over Ap(Ω)Ap(Ω). Moreover, motivated by their significance in topology and the study of mapping class groups, we consider a class of maps, referred to as twists, and examine them in connection with the corresponding Euler–Lagrange equations and investigate various qualitative properties of the resulting solutions, the stationary twists. Particular attention is paid to the special case of the so-called pp-Dirichlet energy, i.e., when View the MathML sourceF(ξ)=p−1|ξ|p.