  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.
