We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations Z O f ðx; DuðxÞÞ dx;with non-standard growth conditions of ðp; qÞ type jzj p pf ðx; zÞpLðjzj q þ 1Þ; poq:In particular, we find that a sufficient condition for minimizers to be regular is q p o nþa n ; where the function f ðx; zÞ is a-Ho¨lder continuous with respect to the x-variable and xAOCR n ; 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 in [43], showing a minimizer with an isolated singularity. r 2004 Elsevier Inc. All rights reserved. MSC: 49J25

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We prove higher integrability for the gradient of vector-valued minimizers of some integral functionals with p − q growth. * We acknowledge the support of MURST 60%, 40% and GNAFA-CNR. 134