Regularity results for vectorial minimizers of a class of degenerate convex integrals Giovanni Cupini - Flavia Giannetti - Raffaella Giova - Antonia Passarelli di Napoli ∗ May 15, 2017 Abstract We establish the higher differentiability and the higher integrability for the gradient of vec- torial minimizers of integral functionals with (p,q)-growth conditions. We assume that the non- homogeneous densities are uniformly convex and have a radial structure, with respect to the gradient variable, only at infinity. The results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand. AMS Classifications. 49N60; 49N99; 35J47. Keywords. Regularity, asymptotically convex, minimizer, (p, q)-growth. 1 Introduction In this paper we study the regularity of vectorial local minimizers of functionals of the form F (u; Ω) :=  Ω f (x,Du) dx, (1.1) where Ω ⊂ R n , n> 2, is a bounded open set, u :Ω → R N , N> 1, is a Sobolev map and f : Ω × R nN → [0, +∞) is a Carath´ eodory function, convex and of class C 2 with respect to the second variable. The main features of the energy densities f (x,ξ ) considered here are the following: • they satisfy the so-called (p,q)-growth conditions • they are uniformly convex only for large values of |ξ | • the partial map x → f (x,ξ ) is possibly discontinuous. ∗ Acknowledgement: The authors thank Matteo Focardi for helpful discussions. Part of this work was carried out while G. Cupini was visiting the Mathematical Department R. Caccioppoli, University of Napoli “Federico II”, and R. Giova and A. Passarelli di Napoli the Mathematical Department of the University of Bologna. These authors thank these Institutions for their warm hospitality. The first author was partially supported by “Progetto inviti 2016” of the Mathematical Department R. Cac- cioppoli, University of Napoli “Federico II”. All the authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). 1