Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients

Abstract

We prove global Lorentz estimates for variable power of the gradient of weak solution to linear elliptic obstacle problems with small partially BMO coefficients over a bounded nonsmooth domain. Here, we assume that the leading coefficients are measurable in one variable and have small BMO semi-norms in the other variables, variable exponents $p(x)$ satisfy log-H\"older continuity, and the boundary of domains are so-called Reifenberg flat. This is a natural outgrowth of the classical Calder\'{o}n-Zygmund estimates to a variable power of the gradient of weak solutions in the scale of Lorentz spaces for such variational inequalities beyond the Lipschitz domain.