  We consider autonomous integral functionals of the form $\mathcal
F[u]:=\int_\Omega f(D u)\,dx$ with $u:\Omega\to\mathbb R^N$ $N\geq1$, where the
convex integrand $f$ satisfies controlled $(p,q)$-growth conditions. We
establish higher gradient integrability and partial regularity for minimizers
of $\mathcal F$ assuming $\frac{q}p<1+\frac2{n-1}$, $n\geq3$. This improves
earlier results valid under the more restrictive assumption
$\frac{q}p<1+\frac2{n}$.
