作者：Luca Capogna，Carlos
页数：10,155页
出版社：高等教育出版社
出版日期：2018
ISBN：9787040503074
电子书格式：pdf/epub/txt
内容简介
几何测度论和调和分析的新近发展带来了相关领域一系列的发展。例如表现为“近似”于欧几里得体积的测度支集的正则性理论获得了深刻的结果，最令人意想不到的是借助于该测度的渐进性，从而刻画了支集的平坦性特征，而这些特征引发了非光滑区域的调和测度的研究中重要的新进展。 本书给出了此领域中最新研究成果的概览和介绍。本书内容基于 Carlos Kenig 于 2000 年在Arkansas 大学的 Arkansas 春季系列讲座中的五讲的讲义，另加以扩充和更新以反映这个领域的快速发展。此外, 还增加了一章平面情形提供历史回顾。 本书包含了背景知识的介绍，便于高年级的研究生和调和分析及几何测度论领域的研究人员理解。
目录
Introduction
Chapter 1.Motivation and statement of the main results
1.Characterization (1)α: Approximation with planes
2.Characterization (2)α: Introducing BMO and VMO
3.Multiplicative vs.additive formulation: Introducing the doubling condition
4.Characterization (1)α and flatness
5.Doubling and asymptotically optimally doubling measures
6.Regularity of a domain and doubling character of its harmonic measure
7.Regularity of a domain and smoothness of its Poisson kernel
Chapter 2.The relation between potential theory and geometry for planar domains
1.Smooth domains
2.Non smooth domains
3.Preliminaries to the proofs of Theorems 2.7 and 2.8
4.Proof of Theorem 2.7
5.Proof of Theorem 2.8
6.Notes
Chapter 3.Preliminary results in potential theory
1.Potential theory in NTA domains
2.A brief review of the real variable theory of weights
3.The spaces BMO and VMO
4.Potential theory in C1 domains
5.Notes
Chapter 4.Reifenberg flat and chord arc domains
1.Geometry of Reifenberg flat domains
2.Small constant chord arc domains
3.Notes
Chapter 5.Further results on Reifenberg fiat and chord arc domains
1.Improved boundary regularity for J-Reifenberg flat domains
2.Approximation and Rellich identity
3.Notes
Chapter 6.From the geometry of a domain to its potential theory
1.Potential theory for Reifenberg domains with vanishing constant
2.Potential theory for vanishing chord arc domains
3.Notes
Chapter 7.From potential theory to the geometry of a domain
1.Asymptotically optimally doubling implies Reifenberg vanishing
2.Back to chord arc domains
3.log k E VMO implies vanishing chord arc; Step I
4.log k E VMO implies vanishing chord arc; Step II
5.Notes
Chapter 8.Higher codimension and further regularity results
1.Notes
Bibliography
Chapter 1.Motivation and statement of the main results
1.Characterization (1)α: Approximation with planes
2.Characterization (2)α: Introducing BMO and VMO
3.Multiplicative vs.additive formulation: Introducing the doubling condition
4.Characterization (1)α and flatness
5.Doubling and asymptotically optimally doubling measures
6.Regularity of a domain and doubling character of its harmonic measure
7.Regularity of a domain and smoothness of its Poisson kernel
Chapter 2.The relation between potential theory and geometry for planar domains
1.Smooth domains
2.Non smooth domains
3.Preliminaries to the proofs of Theorems 2.7 and 2.8
4.Proof of Theorem 2.7
5.Proof of Theorem 2.8
6.Notes
Chapter 3.Preliminary results in potential theory
1.Potential theory in NTA domains
2.A brief review of the real variable theory of weights
3.The spaces BMO and VMO
4.Potential theory in C1 domains
5.Notes
Chapter 4.Reifenberg flat and chord arc domains
1.Geometry of Reifenberg flat domains
2.Small constant chord arc domains
3.Notes
Chapter 5.Further results on Reifenberg fiat and chord arc domains
1.Improved boundary regularity for J-Reifenberg flat domains
2.Approximation and Rellich identity
3.Notes
Chapter 6.From the geometry of a domain to its potential theory
1.Potential theory for Reifenberg domains with vanishing constant
2.Potential theory for vanishing chord arc domains
3.Notes
Chapter 7.From potential theory to the geometry of a domain
1.Asymptotically optimally doubling implies Reifenberg vanishing
2.Back to chord arc domains
3.log k E VMO implies vanishing chord arc; Step I
4.log k E VMO implies vanishing chord arc; Step II
5.Notes
Chapter 8.Higher codimension and further regularity results
1.Notes
Bibliography
