Extremal mappings of finite distortion and the Radon-Riesz property

Abstract

We consider Sobolev mappings $f\in W^{1,q}(Ω,\IC)$, $1<q<\infty$, between planar domains $Ω\subset \IC$. We analyse the Radon-Riesz property for convex functionals of the form \[f\mapsto \int_ΩΦ(|Df(z)|,J(z,f)) \; dz \] and show that under certain criteria, which hold in important cases, weak convergence in $W_{loc}^{1,q}(\Omega )$ of (for instance) a minimising sequence can be improved to strong convergence. This finds important applications in the minimisation problems for mappings of finite distortion and the $L^p$ and $Exp$\,-Teichmüller theories.