In the 1870s, Carl Neumann proposed the so-called method of the arithmetic mean for solving the Dirichlet problem on convex domains. Neumann’s approach was considered at the time to be a reliable existence proof, following Weierstrass’s criticism of the Dirichlet principle. However, in 1937 H. Lebesgue pointed out a serious gap in Neumann’s proof. Curiously, the erroneous argument once again involved confusion between the notions of infimum and minimum. The objective of this paper is to show that Lebesgue’s sharp criticism of Neumann’s proof was only partially justified.

中文翻译：

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勒贝格对卡尔诺依曼势论方法的批判

1870年代，卡尔·诺依曼提出了所谓的算术平均法来求解凸域上的狄利克雷问题。在魏尔斯特拉斯对狄利克雷原理的批评之后，诺依曼的方法在当时被认为是可靠的存在证明。然而，在 1937 年，H. Lebesgue 指出了诺依曼证明中的一个严重漏洞。奇怪的是，这个错误的论证再次涉及对下限和最小值概念的混淆。本文的目的是表明勒贝格对诺依曼证明的尖锐批评只是部分合理的。