Optimal doubling, reifenberg flatness and operators of p-laplace type

• 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−1A(x,η/|η|) whenever x∈Rn and η∈Rn∖{0}. Let Ω⊂Rn be a bounded domain, let D be a compact subset of Ω. We say that is the A-capacitary function for D in Ω if on D, on ∂Ω in the sense of and in Ω∖D in the weak sense. We extend to Rn∖Ω by putting on Rn∖Ω. Then there exists a unique finite positive Borel measure on Rn, with support in ∂Ω, such that In this paper, we prove that if Ω is Reifenberg flat with vanishing constant, then for every τ, 0<τ≤1. In particular, we prove that is an asymptotically optimal doubling measure on ∂Ω.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Reifenberg flat domain

Reifenberg flat domain with vanishing constant

p-harmonic function

A-harmonic function

Variable coefficients

Doubling measure

Asymptotically optimal doubling measure

MATHEMATICS

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