边界理论、外逼近，与分形上的微分方程

项目名称： 边界理论、外逼近，与分形上的微分方程

项目编号： No.11271122

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 倪思敏

作者单位： 湖南师范大学

项目金额： 60万元

中文摘要： 本项目将循概率和分析两途径研究分形上的调和函数、狄氏形、拉普拉斯算子与微分方程。我们着重于非后临界有限，特别是有重叠的自相似集。一直以来,分形分析主要采取两方法：一是通过构造布朗运动的概率方法，二是Kigami内逼近的分析方法。然而，目前的理论只能应用在几类没重叠的自相似集上。 边界理论为分形研究提供了另一概率途径。我们将用该丰富的理论推广近期有关Martin边界与调和函数的结果, 并研究拉普拉斯算子。另一方面, 我们将通过分析途径研究高维有界开集上，由自相似测度定义的拉普拉斯算子。我们提议通过热核估计，解决波速问题。在此基础上，我们将结合外逼近的方法，定义自相似集上的拉普拉斯算子。 本项目的两途径是互补且非传统的。除了上述的理论和方法，我们还将利用有限型及弱分离两条件。我们的目标是在有重叠的自相似集上，建立分形分析的基本理论, 为进一步研究分形上的物理现象和分形的应用打下基础。

中文关键词： 拉普拉斯算子；热核估计；波动方程；外逼近；有限型条件

英文摘要： This project adopts both probabilistic and analytic approaches to study harmonic functions, Dirichlet forms, Laplacians and differential equations on fractals, especially self-similar sets that do not satisfy the post critically finite (p.c.f.) condition and those that have overlaps. Traditionally, there are two major methods in analysis on fractals: the probabilistic methods that constructs of Brownian motions, and the direct analytic method developed by Kigami that uses inner approximations. However, these approaches can only be applied to several classes of self-similar sets without overlaps. Boundary theory provides an alternative probabilistic approach for studying fractals. We will use this rich theory to extend recent results on Martin boundaries and harmonic functions, and define the Laplacian. On the other hand, we will use an analytic approach to study higher dimensional fractal Laplacians defined on bounded open sets by self-similar measures with overlaps. We propose to solve the wave propagation speed problem by first obtaining heat kernel estimates. Then we will use the theory developed on bounded open sets by our analytic approach, along with the idea of outer approximation, to construct Laplacians on self-similar sets. Our probabilistic and analytic approaches are complementary to each other a

英文关键词： Laplacian；heat kernel estimate；wave equation；outer approximation；finite type condition