In this paper, we consider equations of p -Laplace type of the form ∇ ⋅ A ( x , ∇ u ) = 0 . Concerning A we assume, for p ∈ ( 1 , ∞ ) fixed, an appropriate ellipticity type condition, Hölder continuity in x and that A ( x , η ) = | η | p − 1 A ( x , η / | η | ) whenever x ∈ R n and η ∈ R n ∖ { 0 } . Let Ω ⊂ R n be a bounded domain, let D be a compact subset of Ω . We say that u ˆ = u ˆ p , D , Ω is the A -capacitary function for D in Ω if u ˆ ≡ 1 on D , u ˆ ≡ 0 on ∂ Ω in the sense of W 0 1 , p ( Ω ) and ∇ ⋅ A ( x , ∇ u ˆ ) = 0 in Ω ∖ D in the weak sense. We extend u ˆ to R n ∖ Ω by putting u ˆ ≡ 0 on R n ∖ Ω . Then there exists a unique finite positive Borel measure μ ˆ on R n , with support in ∂ Ω , such that ∫ 〈 A ( x , ∇ u ˆ ) , ∇ ϕ 〉 d x = − ∫ ϕ d μ ˆ whenever ϕ ∈ C 0 ∞ ( R n ∖ D ) . In this paper, we prove that if Ω is Reifenberg flat with vanishing constant, then lim r → 0 inf w ∈ ∂ Ω μ ˆ ( B ( w , τ r ) ) μ ˆ ( B ( w , r ) ) = lim r → 0 sup w ∈ ∂ Ω μ ˆ ( B ( w , τ r ) ) μ ˆ ( B ( w , r ) ) = τ n − 1 , for every τ , 0 < τ ≤ 1 . In particular, we prove that μ ˆ is an asymptotically optimal doubling measure on ∂ Ω .