Extremal mappings of finite distortion and the Radon–Riesz property

Abstract

We consider Sobolev mappings $f\in W^{1,q}(\Omega,\mathbb{C})$, $1\<q<\infty$, between planar domains $\Omega\subset \mathbb{C}$. We analyse the Radon–Riesz property for polyconvex functionals of the form $$ f\mapsto \int\_\Omega \Phi(|Df(z)|,J(z,f)) , dz $$ and show that under certain criteria, which hold in important cases, weak convergence in $W\_{\mathrm{loc}}^{1,q}(\Omega)$ 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.