GRADIENT REGULARITY FOR MINIMIZERS OF FUNCTIONALS UNDER p − q SUBQUADRATIC GROWTH F. LEONETTI, E. MASCOLO & F. SIEPE Sunto: Si prova la maggior sommabilit`adel gradiente dei minimi locali di funzionali integrali della forma  Ω f (Du)dx, dove f soddisfa l’ipotesi di crescita |ξ | p − c 1 ≤ f (ξ ) ≤ c(1 + |ξ | q ) e si assume che 1 <p<q ≤ 2. 1. Introduction. Let us consider the functional F (u, Ω) =  Ω f (Du(x))dx (1.1) where Ω is a bounded open set in R n , n ≥ 2, Du is the gradient of a vector valued function u :Ω → R N , N ≥ 1, and f : R nN → R. In this paper we study the local regularity of minimizers of F . In particular, we consider the case in which the integrand function f satisfies the p − q growth condition |ξ | p − c 1 ≤ f (ξ ) ≤ c(1 + |ξ | q ) (1.2) with p<q. The regularity properties of minimizers, under assumption (1.2), has been in- tensely studied in the last years. F. Leonetti: Dipartimento di Matematica Pura ed Applicata, Universit`a di L’Aquila - 67100 L’Aquila, Italy. E-Mail: leonetti@univaq.it E. Mascolo: Dipartimento di Matematica ”U. Dini”, Universit` a di Firenze, Viale Morgagni 67/A - 50134 Firenze, Italy. E-Mail: mascolo@udini.math.unifi.it F. Siepe: Dipartimento di Matematica ”U. Dini”, Universit` a di Firenze, Viale Morgagni 67/A - 50134 Firenze, Italy. E-Mail: siepe@udini.math.unifi.it Key words and phrases: minimizers, non-standard growth conditions, p - q growth, higher inte- grability. Mathematics subjects classification (Amer. Math. Soc.): 49N60, 35J60. We acknowledge the support of MURST (40%, 60%) and GNAFA-CNR. 1