Sharp regularity for functionals with (q,p) growth

We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations f(x; u;Du) dx; with non-standard growth conditions of (p; q) type |z |p< f(x; z) < L(| z| ^q +1); p < q: In particular,we prove that a sufficient condition for minimizers to be regular is q/p<(n+\alpha)/n ; where the function f(x; z) is Ho lder continuous with respect to the x-variable. This condition is also sharp. We include results in the setting of Orlicz spaces; moreover,we treat certain relaxed functionals too. Finally,we address a problem posed by Marcellini, showing a minimizer with an isolated singularity.