对内外场的对数势，第二版

English | 2024 | ISBN: 978-3-031-65133-5 | 598 pages | PDF EPUB (True) | 11 MB

This is the second edition of an influential monograph on logarithmic potentials with external fields, incorporating some of the numerous advancements made since the initial publication.

As the title implies, the book expands the classical theory of logarithmic potentials to encompass scenarios involving an external field. This external field manifests as a weight function in problems dealing with energy minimization and its associated equilibria. These weighted energies arise in diverse applications such as the study of electrostatics problems, orthogonal polynomials, approximation by polynomials and rational functions, as well as tools for analyzing the asymptotic behavior of eigenvalues for random matrices, all of which are explored in the book. The theory delves into diverse properties of the extremal measure and its logarithmic potentials, paving the way for various numerical methods.

This new, updated edition has been thoroughly revised and is reorganized into three parts, Fundamentals, Applications and Generalizations, followed by the Appendices. Additions to the new edition include

new material on the following topics: analytic and C² weights, differential and integral formulae for equilibrium measures, constrained energy problems, vector equilibrium problems, and a probabilistic approach to balayage and harmonic measures; a new chapter entitled Classical Logarithmic Potential Theory, which conveniently summarizes the main results for logarithmic potentials without external fields; several new proofs and sharpened forms of some main theorems; expanded bibliographic and historical notes with dozens of additional references. Aimed at researchers and students studying extremal problems and their applications, particularly those arising from minimizing specific integrals in the presence of an external field, this book assumes a firm grasp of fundamental real and complex analysis. It meticulously develops classical logarithmic potential theory alongside the more comprehensive weighted theory.

这是关于有外场对数位势的 influential 手册的第二版，包含了自首次出版以来众多进步的若干。 正如书名所示，该书扩展了经典位势理论以涵盖涉及外场的情况。这种外部场表现为问题中的权重函数。这些加权能量在诸如电容器的研究、正交多项式的研究、多项式和有理函数的逼近，以及随机矩阵特征值渐近行为分析等方面都有应用。书中探讨了许多关于极值测度及其位势的特性，为各种数值方法铺平了道路。 新修订版已全面更新并重新组织成三个部分：基础、应用与推广，接着是附录。 新增内容包括： - 分析和 \(C^2\) 权重的新材料；均衡测度的微分和积分表示式；约束能量问题；向量均衡问题；以及概率方法用于漂移和调和测度； - 一本新章节，名为经典位势理论，方便总结无外场情况下的主要结果； - 几个新的证明以及一些主要定理的新形式； - 扩展的参考文献和历史注释，包含数十个额外引用。 本书旨在针对研究极值问题及其应用的研究人员和学生。特别是那些在外部场存在的情况下，致力于最小化特定积分的问题。该书假设读者具备了基本的实分析和复分析知识。它详细发展了经典位势理论，并将权重理论进行了全面探讨。

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