On the regularity of very weak solutions for linear elliptic equations in divergence form

In this paper we consider a linear elliptic equation in divergence form $$D_j(a_{ij}(x)D_iu) = 0 in Ω.$$ (0.1) Assuming the coefficients a_{ij} in W^{1,n}(Ω) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution u ∈ L^{n′} (Ω) of (0.1) is actually a weak solution in W^{1,2} (Ω)