Regularity of very weak solutions for elliptic equation of divergence form

In this paper, we study the local regularity of very weak solution u ∈ L1 loc(Ω) of the elliptic equation Dj (aij (x)Diu) = 0. Using the bootstrap argument and the difference quotient method, we obtain that if aij ∈ C 0,1 loc (Ω), then u ∈W 2,p loc (Ω) for any p <∞. © 2011 Elsevier Inc. All rights reserved.