论文标题
$ 4N $ -CARPETS的阻力缩放
Resistance Scaling on $4N$-Carpets
论文作者
论文摘要
$ 4N $地毯是一类无限分配的自相似分形，上面有大量的对称性。对于$ 4N $ -CARPET $ f $，令$ \ {f_n \} _ {n \ geq 0} $是带有$ \ cap_nf_n = f $的紧凑型预犯近似值的自然减小序列。 On each $F_n$, let $\mathcal{E}(u, v) = \int_{F_N} \nabla u \cdot \nabla v \, dx$ be the classical Dirichlet form and $u_n$ be the unique harmonic function on $F_n$ satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary.使用Barlow和Bass（1990）介绍的方法，我们证明了以下形式的电阻估计：$ρ=ρ（n）> 1 $ $，因此$ \ nathcal {e}（u_n，u_n，u_n）ρ^{n} $在上方和下方由独立于$ n $ $ n $的积极常数限制。这些估计值对$ f $上的dirichlet表单的存在和缩放属性具有影响。
The $4N$ carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a $4N$-carpet $F$, let $\{F_n\}_{n \geq 0}$ be the natural decreasing sequence of compact pre-fractal approximations with $\cap_nF_n=F$. On each $F_n$, let $\mathcal{E}(u, v) = \int_{F_N} \nabla u \cdot \nabla v \, dx$ be the classical Dirichlet form and $u_n$ be the unique harmonic function on $F_n$ 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 $\mathcal{E}(u_n, u_n)ρ^{n}$ 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 $F$.
