Abstract

During almost two centuries after the Gauss’ formulation of the Dirichlet problem for Laplace equation, many famous mathematicians devoted their studies to this subject and to its various generalizations. Many interesting and important results have been obtained, which become already classical ones. Our paper is an extended presentation of the author’s talk on the international conference dedicate to the century of V. S. Vladimirov birthday. Its main content is the review of the results in that direction, including the proves of new statements and discussion of unsolved problems. Our goal is to convince readers that, in this “principal” problem of mathematical physics, we know far from everything even about the case of linear equation. There are many interesting and important unsolved problems in that direction.

中文翻译：

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关于狄利克雷问题

摘要

在高斯提出拉普拉斯方程的狄利克雷问题后的近两个世纪里，许多著名的数学家致力于这一主题及其各种推广的研究。获得了许多有趣且重要的成果，这些成果已经成为经典。我们的论文是作者在纪念弗拉基米罗夫诞辰世纪国际会议上的演讲的延伸介绍。其主要内容是对该方向的成果的回顾，包括新陈述的证明和未解决问题的讨论。我们的目标是让读者相信，在这个数学物理的“主要”问题中，我们远非了解一切，甚至连线性方程的情况也是如此。在这个方向上还有许多有趣且重要的未解决问题。