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

Abstract

Let Ω ⊂ R n be a bounded Lipschitz domain, F : R n × n → R a suitably quasiconvex integrand and consider the energy functional F [ u , Ω ] ≔ ∫ Ω F ( ∇ u ) , over the space of measure preserving maps A p ( Ω ) = { u ∈ W 1 , p ( Ω , R n ) : u | ∂ Ω = x , det ∇ u = 1 a . e . in Ω } . In this paper we discuss the question of existence of multiple strong local minimizers for F over A p ( 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 p -Dirichlet energy, i.e., when F ( ξ ) = p − 1 | ξ | p .