Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains

Abstract

In a uniform domain Ω, we present a certain reverse mean value inequality and a Harnack type inequality for positive superharmonic functions satisfying a nonlinear inequality -Δu(x) ≤ cδ_Ω(x)^-αu(x)^p for x ∈ Ω, where c > 0, α ≥ 0 and p > 1 and δ_Ω(x) is the distance from a point x to the boundary of Ω. These are established by refining a boundary growth estimate obtained in our previous paper (2008). Also, we apply them to show the existence of nontangential limits of quotients of such functions and to give an extension of a certain minimum principle studied by Dahlberg (1976).