L p -estimates for the solutions of second order elliptic equations

Summary We investigate the homogeneous Dirichlet problem in H^2,p for a second order elliptic partial differential equation in nondivergence form Lu=f in the case in which the leading coefficients of L belong to H^1,n(), Rⁿ. We prove that if p belongs to a suitable neighbourhood of 2, then the above problem, has a unique solution u satisfying D²u_p Cf_p; furthermore, if f H^k,p, k=1,2, ..., and the coefficients of L satisfy some natural conditions, then the solution satisfies .Lavoro eseguito nell'ambito del gruppi 40% e 60% del M.P.I.