Extremal mappings of finite distortion and the Radon-Riesz property

TL;DR

This paper investigates the Radon-Riesz property for convex functionals of Sobolev mappings, showing conditions under which weak convergence implies strong convergence, with applications to finite distortion mappings and Teichmüller theories.

Contribution

It establishes criteria for the Radon-Riesz property in Sobolev mappings, enhancing understanding of convergence in minimization problems related to finite distortion.

Findings

Weak convergence can be upgraded to strong convergence under certain criteria.

The results apply to minimization problems in finite distortion and Teichmüller theories.

Provides new insights into the structure of Sobolev mappings and their convergence properties.

Abstract

We consider Sobolev mappings $f\in W^{1,q}(\Omega,\IC)$, $1<q<\infty$, between planar domains $\Omega\subset \IC$. We analyse the Radon-Riesz property for convex 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 $W_{loc}^{1,q}(\Omega)$ 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 $Exp$\,-Teichm\"uller theories.