Some remarks on variational problems for functionals with $L\ln L$ growth

Regularity for minimizers of the functional $\int_\Omega|\nabla v|\ln(1+|\nabla v|)\,dx$ on a set of vector-valued functions $v\colon\Omega\subset\mathbb R^n\to\mathbb R^n$, taking prescribed values on the boundary $\partial\Omega$, is studied. It is shown that solution of the dual variational problem belong to the class $W^1_{2,\mathrm{loc}}$. In the case $n=2$ a higher integrability for minimizers of the direct variational problem is proved. Bibliography: 5 titles.