关于分形域上能量形式的一个注记
A note on energy forms on fractal domains
摘要：
J. Kigami通过允许在具有分形特征的图上构造具有拉普拉斯算子相同性质的算子，奠定了现在称为分形分析的基础。Sierpinski垫片就是最著名的例子。从那时起，它被R. S.哈茨拉普拉斯算子是通过一个弱公式获得的，该公式是通过狄利克雷形式获得的，该狄利克雷形式是通过对一系列收敛于所考虑的域的图进行归纳而建立的。正是这些狄利克雷形式，使人们能够获得能量形式在这个领域。然而，事情并没有那么简单。如果，对于像谢尔宾斯基垫片这样的区域，拉普拉斯算子是以一种非常自然的方式获得的，那么我们必须记住，狄利克雷形式只依赖于区域的拓扑结构，而不是它的几何结构。这意味着，如果一个人的目标是建立一个分形域上的拉普拉斯算子，其拓扑结构是相同的，例如，一个线段，一个人必须找到一种方法来考虑一个非常具体的几何形状。我们在研究维尔斯特拉斯函数的图形时遇到了这个问题。因此，解决方案是考虑比经典形式更复杂的能量形式，通过归一化常数，不仅可以承载拓扑结构，而且可以承载非常特定的几何形状，从现在开始，我们将称为W曲线。我们目前的目标是研究能量形式和几何之间的联系。我们选择考虑分形曲线，具体地说，Sierpienski箭头曲线，其极限是Sierpienski垫片。是否得到与三角形相同的拉普拉斯算子？这个问题似乎值得研究。
英文摘要：
J. Kigami has laid the foundations of what is now known as analysis on fractals, by allowing the construction of an operator of the same nature of the Laplacian, defined locally, on graphs having a fractal character. The Sierpinski gasket stands out of the best known example. It has, since then, been taken up, developed and popularized by R. S. Strichartz. The Laplacian is obtained through a weak formulation, obtained by means of Dirichlet forms, built by induction on a sequence of graphs that converges towards the considered domain. It is these Dirichlet forms that enable one to obtain energy forms on this domain. Yet, things are not that simple. If, for domains like the Sierpinski gasket, the Laplacian is obtained in a quite natural way, one must bear in mind that Dirichlet forms solely depend on the topology of the domain, and not of its geometry. Which means that, if one aims at building a Laplacian on a fractal domain, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account a very specific geometry. We came across that problem in our work on the graph of the Weierstrass function The solution was thus to consider energy forms more sophisticated than classical ones, by means of normalization constants that could, not only bear the topology, but, also, the very specific geometry of, from now on, we will call W-curves. We presently aim at investigating the links between energy forms and geometry. We have chosen to consider fractal curves, specifically, the Sierpiński arrowhead curve, the limit of which is the Sierpiński gasket. Does one obtain the same Laplacian as for the triangle ? The question appears as worth to be investigated.
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