通过应用介绍位势论
Introduction to Potential Theory via Applications
摘要：
介绍了与下调和函数和位势有关的基本概念，主要是复平面上的情形，并证明了Riesz分解定理。除了理论的基本事实之外，我们稍微偏离了通常的阐述路径，并在应用的同时引入了更多的概念。我们详细讨论了狄利克雷问题，并通过考虑调和测度及其与布朗运动的关系，说明了位势论与概率的关系.此外，格林函数的介绍和应用的多项式的增长。平衡测度的产生是由于它们在物理学中的最初发展，最后简要讨论了容量及其与豪斯多夫测度的关系。我们希望熟悉实分析、复分析、测度论和某些概率论的主要内容的读者能从这些笔记中受益。没有新的结果提出，但我们希望，风格的介绍，使读者能够迅速理解的基本思想的位势论，以及它如何可以在不同的情况下使用。这些笔记也可以用于位势论的短期课程。因此，附录中描述了所需的先决条件。在论述和细节可以找到的地方都给出了参考资料;粗略地说，熟悉真实和复分析的基本基础应该足以在没有任何背景阅读的情况下继续进行。
英文摘要：
We introduce the basic concepts related to subharmonic functions and potentials, mainly for the case of the complex plane and prove the Riesz decomposition theorem. Beyond the elementary facts of the theory we deviate slightly from the usual path of exposition and introduce further concepts alongside with applications. We cover the Dirichlet problem in detail and illustrate the relations between potential theory and probability by considering harmonic measure and its relation to Brownian motion. Furthermore Green's function is introduced and an application to growth of polynomials is given. Equilibrium measures are motivated by their original development in physics and we end with a brief discussion of capacity and its relation to Hausdorff measure. We hope that the reader, who is familiar with the main elements of real analysis, complex analysis, measure theory and some probability theory benefits from these notes. No new results are presented but we hope that the style of presentation enables the reader to understand quickly the basic ideas of potential theory and how it can be used in different contexts. The notes can also be used for a short course on potential theory. Therefore the required prerequisites are described in the Appendix. References are given where expositions and details can be found; roughly speaking, familiarity with the basic foundations of real and complex analysis should suffice to proceed without any background reading.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
