Potential Theory on Sierpiński Carpets
Potential Theory on Sierpiński Carpets
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谢尔宾斯基地毯的潜力理论
DOI：
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发表时间：
2018
影响因子：
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通讯作者：
Dimitrios Ntalampekos
中科院分区：
文献类型：
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作者：
Dimitrios Ntalampekos
Author（s）：Ntalampekos，Dimitrios|指导：Bonk，Mario|摘要：这项研究的动机是分形集的几何学的研究，并集中在统一化问题：集的转换为规范集，使用地图，保持在某种意义上的几何。更具体地说，主要的问题是解决的平面Sierpinski地毯的均匀化平方Sierpinski地毯，使用地毯的潜在理论的方法。我们介绍了一个离散的概念Sobolev空间Sierpinski地毯，并使用此定义调和函数。我们的方法不同于度量空间中的经典势理论方法，因为它考虑了包含地毯的环境空间。我们证明的基本性质，如Dirichlet问题的解的存在性和唯一性，刘维定理，Harnack不等式，强极大值原理，调和函数的等度连续性。然后利用调和函数的概念证明了Sierpinski地毯的一个均匀化结果。即证明了每一个周边圆盘为均匀胖、均匀拟球的平面Sierpinski地毯都可以映射到一个具有保持地毯模的映射的正方形Sierpinski地毯.如果把关于圆周圆的假设加强到均匀相对分离的均匀拟圆，则该映射是拟对称的。一致化映射的真实的部分是某个Dirichlet型问题的解。然后使用Rajala开发的方法构造该映射的调和共轭。
Author(s): Ntalampekos, Dimitrios | Advisor(s): Bonk, Mario | Abstract: This research is motivated by the study of the geometry of fractal sets and is focused on uniformization problems: transformation of sets to canonical sets, using maps that preserve the geometry in some sense. More specifically, the main question addressed is the uniformization of planar Sierpinski carpets by square Sierpinski carpets, using methods of potential theory on carpets.We first develop a potential theory and study harmonic functions on planar Sierpinski carpets. We introduce a discrete notion of Sobolev spaces on Sierpinski carpets and use this to define harmonic functions. Our approach differs from the classical approach of potential theory in metric spaces because it takes the ambient space that contains the carpet into account. We prove basic properties such as the existence and uniqueness of the solution to the Dirichlet problem, Liouville's theorem, Harnack's inequality, strong maximum principle, and equicontinuity of harmonic functions. Then we utilize this notion of harmonic functions to prove a uniformization result for Sierpinski carpets. Namely, it is proved that every planar Sierpinski carpet whose peripheral disks are uniformly fat, uniform quasiballs can be mapped to a square Sierpinski carpet with a map that preserves carpet modulus. If the assumptions on the peripheral circles are strengthened to uniformly relatively separated, uniform quasicircles, then the map is a quasisymmetry. The real part of the uniformizing map is the solution of a certain Dirichlet-type problem. Then a harmonic conjugate of that map is constructed using the methods developed by Rajala.