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2020 | Book

Introduction Read chapter Read first chapter

Potential Theory on Sierpiński Carpets

With Applications to Uniformization

Author: Dimitrios Ntalampekos

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This self-contained book lays the foundations for a systematic understanding of potential theoretic and uniformization problems on fractal Sierpiński carpets, and proposes a theory based on the latest developments in the field of analysis on metric spaces. The first part focuses on the development of an innovative theory of harmonic functions that is suitable for Sierpiński carpets but differs from the classical approach of potential theory in metric spaces. The second part describes how this theory is utilized to prove a uniformization result for Sierpiński carpets. This book is intended for researchers in the fields of potential theory, quasiconformal geometry, geometric group theory, complex dynamics, geometric function theory and PDEs.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
One of the main problems in the field of Analysis on Metric Spaces is to find geometric conditions on a given metric space under which the space can be transformed to a “canonical” space with a map that preserves the geometry. In other words, we wish to uniformize the metric space by a canonical space. For example, the Riemann mapping theorem gives a conformal map from any simply connected proper subregion of the plane onto the unit disk, which is the canonical space in this case.
Dimitrios Ntalampekos
Chapter 2. Harmonic Functions on Sierpiński Carpets
Abstract
In this chapter we introduce and study notions of Sobolev spaces and harmonic functions on Sierpiński carpets. We briefly describe here some of the applications of carpet-harmonic functions, and then the organization of the current chapter.
Dimitrios Ntalampekos
Chapter 3. Uniformization of Sierpiński Carpets by Square Carpets
Abstract
In this chapter we prove a uniformization result for planar Sierpiński carpets by square Sierpiński carpets, by minimizing some kind of energy. For the convenience of the reader, we include here some definitions, some of which are also given in Chap. 2. We will also point out, whenever necessary, any discrepancies in the notation between the two chapters. However, for the most part, this chapter is independent of Chap. 2 and we will only use certain results from there that we quote again here.
Dimitrios Ntalampekos
Backmatter
Metadata
Title
Potential Theory on Sierpiński Carpets
Author
Dimitrios Ntalampekos
Copyright Year
2020
Electronic ISBN
978-3-030-50805-0
Print ISBN
978-3-030-50804-3
DOI
https://doi.org/10.1007/978-3-030-50805-0

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