On the Homogenization of Quasilinear Divergence Structure Operators (*). ~ . FUSCO - G. MOSCARIELLO Summary. - We study the homogenization o/ second order quasilinear operators o] the ]orm A~u =-- diva(X,u, Du) in Sobolev spaces H~,~ (p > 1). An explicit formula o] the homogenized operator is given. 1. - Introduction... In this paper we study the homogenization of a family of quasilinear operators (1.1) u~//~'~(~) p>l, where a($, u, ~) is periodic in $ and verifies suitable growth conditions, e > 0. Indeed we prove that the solutions u, of the problems A,u = l (1.2) / converge in the weak topology of H10'~(TJ) to a function Uo which is the solution of the problem: -- div b(u, D~t) = ] e H~,'(~) and a(x/e, u,, Du~) converge to b(uo, Duo) in the weak topology of L ~', with p'= = p/(p- 1). Moreover the matrix b(u, ~) is given by an explicit formula. (*) Entrata in Redazione 1'8 settembre 1984. (*) Indirizzo degli A.: Dipartimento di Matematica, UniversitY, Via Mezzocannone 8, 80134 Napoli.