Optimal doubling, Reifenberg flatness and operators of pp-Laplace type

In this paper, we consider equations of pp-Laplace type of the form ∇⋅A(x,∇u)=0∇⋅A(x,∇u)=0. Concerning AA we assume, for p∈(1,∞)p∈(1,∞) fixed, an appropriate ellipticity type condition, Hölder continuity in xx and that A(x,η)=|η|p−1A(x,η/|η|)A(x,η)=|η|p−1A(x,η/|η|) whenever x∈Rnx∈Rn and η∈Rn∖{0}η∈Rn∖{0}. Let Ω⊂RnΩ⊂Rn be a bounded domain, let DD be a compact subset of ΩΩ. We say that uˆ=uˆp,D,Ω is the AA-capacitary function for DD in ΩΩ if uˆ≡1 on DD, uˆ≡0 on ∂Ω∂Ω in the sense of W01,p(Ω) and ∇⋅A(x,∇uˆ)=0 in Ω∖DΩ∖D in the weak sense. We extend uˆ to Rn∖ΩRn∖Ω by putting uˆ≡0 on Rn∖ΩRn∖Ω. Then there exists a unique finite positive Borel measure μˆ on RnRn, with support in ∂Ω∂Ω, such that ∫〈A(x,∇uˆ),∇ϕ〉dx=−∫ϕdμˆwhenever ϕ∈C0∞(Rn∖D). In this paper, we prove that if ΩΩ is Reifenberg flat with vanishing constant, then limr→0infw∈∂Ωμˆ(B(w,τr))μˆ(B(w,r))=limr→0supw∈∂Ωμˆ(B(w,τr))μˆ(B(w,r))=τn−1, for every ττ, 0<τ≤10<τ≤1. In particular, we prove that μˆ is an asymptotically optimal doubling measure on ∂Ω∂Ω.