Removable Singularities for Lu=Ψ(u) and Orlicz Capacities

Abstract

Suppose L is a second order elliptic differential operator in Rd and let α>1. Baras and Pierre have proved in 1984 that Γ is removable for Lu=uα if and only if its Bessel capacity Cap2, α′(Γ)=0. We extend this result to a general equation Lu=Ψ(u) where Ψ(u) is an increasing convex function subject to Δ2 and ∇2 conditions. Namely, we prove that Γ is removable for Lu=Ψ(u) if and only if its Orlicz capacity is zero, that is, the integral ∫BdxΨ(∫Γ|x−y|2−dν(dy)) is equal to 0 or ∞ for every measure ν concentrated on Γ, where B stands for any ball containing Γ.