有限分歧胞结构分形上的调和坐标
Harmonic coordinates on fractals with finitely ramified cell structure
摘要：
我们定义了一个具有分枝胞腔结构的集合，它是p.c.f.的推广。自相似集介绍了Kigami和fractafolds介绍了Shakehartz。在一般情况下，我们甚至不假设局部自相似性，并允许可数许多细胞连接在每个结点。证明了如果Kigami的阻力形式满足一定的假设，则存在一个弱黎曼度量，使得能量可以表示为弱梯度关于能量测度的范数平方的积分.进一步证明了如果这样的集合在调和坐标下可以同胚表示，则对于光滑函数，弱梯度可以用通常梯度代替。我们还证明了一个简单的公式的能量测度拉普拉斯在调和坐标。
英文摘要：
We define sets with finitely ramified cell structure, which are generalizations of p.c.f. self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.
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