ftp ejde.math.txstate.edu ELLIPTIC PROBLEMS ON THE SPACE OF WEIGHTED WITH THE DISTANCE TO THE BOUNDARY INTEGRABLE FUNCTIONS REVISITED

Abstract. We revisit the regularity of very weak solution to second-order elliptic equations Lu = f in Ω with u = 0 on ∂Ω for f ∈ L1(Ω, δ), δ(x) the distance to the boundary ∂Ω. While doing this, we extend our previous results (and many others in the literature) by allowing the presence of distributions f+g which are more general than Radon measures (more precisely with g in the dual of suitable Lorentz-Sobolev spaces) and by making weaker assumptions on the coefficients of L. One of the new tools is a Hardy type inequality developed recently by the second author. Applications to the study of the gradient of solutions of some singular semilinear equations are also given