Extremal mappings of finite distortion and the Radon–Riesz property

Abstract

We consider Sobolev mappings f ∈ W (Ω,C), 1 < q < ∞, between planar domains Ω ⊂ C. We analyse the Radon-Riesz property for convex functionals of the form f 7→ ∫ Ω Φ(|Df(z)|, J(z, f)) dz and show that under certain criteria, which hold in important cases, weak convergence in W 1,q loc (Ω) 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 and Exp -Teichmüller theories.