  We mainly discuss superquadratic minimization problems for splitting-type
variational integrals on a bounded Lipschitz domain $\Omega \subset
\mathbb{R}^2$ and prove higher integrability of the gradient up to the boundary
by incorporating an appropriate weight-function measuring the distance of the
solution to the boundary data. As a corollary, the local H\"older coefficient
with respect to some improved H\"older continuity is quantified in terms of the
function ${\rm dist}(\cdot,\partial \Omega)$. The results are extended to
anisotropic problems without splitting structure under natural growth and
ellipticity conditions. In both cases we argue with variants of Caccioppoli's
inequality involving small weights
