Regularity results for the gradient of solutions of a class of linear elliptic systems with L1,λL1,λ data

We prove that if a vector-function ff belongs to the Morrey space L1,λ(Ω,RN)L1,λ(Ω,RN), with Ω⊂RnΩ⊂Rn, n≥3n≥3, N≥2N≥2, λ∈]0,n−2]λ∈]0,n−2], and uu is the solution of the system {−Di(Aij(x)Dju)=fin Ωu∈W01,1(Ω,RN)then DuDu belongs to the space Lq,n−q(n−λ−1)(Ω,RnN)Lq,n−q(n−λ−1)(Ω,RnN), for any q∈[1,nn−1[, provided the matrix of bounded measurable coefficients (Aij)(Aij) has sufficiently small dispersion of the eigenvalues.