2020/11/10 by Claire Canner, Canner, Claire, Christopher Hayes +7
Mathematics · Physics and Astronomy · #28A80 #31C15 #31E05 #331C25 #60J65 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Probability (math.PR) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2011.10662
openalex publication_date 2020/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The 4N carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4N-carpet F, let \Fn\n ≥ 0 be the natural decreasing sequence of compact pre-fractal approximations with ∩nFn=F. On each Fn, let E(u, v) = ∫FN ∇ u ⋅ ∇ v dx be the classical Dirichlet form and un be the unique harmonic function on Fn satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by Barlow and Bass (1990), we prove a resistance estimate of the following form: there is ρ=ρ(N) > 1 such that E(un, un)ρn is bounded above and below by positive constants independent of n. Such estimates have implications for the existence and scaling properties of Dirichlet forms on F.