Regularity for minimizers of functionals with nonstandard growth by   $A$-harmonic approximation
  We prove partial regularity for minimizers of quasiconvex functionals of the type $\int_\Omega f(x,Du) dx$ with $p(x)$ growth with respect to the second variable. The proof is direct and uses a method of $A$-harmonic approximation. 
J. Habermann, A. Zatorska-Goldstein