Abstract In this paper, we consider eigenvalue problems of a second-order measure differential equation. We first study some general theories regarding the completely continuity of eigenvalues, the variational characterizations of eigenvalues, and the oscillating properties of eigenfunctions. Then, we solve the following minimization problem: when the m-th Neumann eigenvalue is given, to find explicitly what measures will have the minimal total variation. As applications of this minimization problem, we will solve some extremal problems of eigenvalues and construct some optimal classes of non-degenerate measures for the second-order measure differential equations.

中文翻译：

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关于二阶测度微分方程的特征值和测度的最小化

摘要 本文研究二阶测度微分方程的特征值问题。我们首先研究了一些关于特征值的完全连续性、特征值的变分表征和特征函数的振荡性质的一般理论。然后，我们解决以下最小化问题：当给定第 m 个 Neumann 特征值时，明确找出哪些度量将具有最小的总变异。作为这个最小化问题的应用，我们将解决一些特征值的极值问题，并为二阶测度微分方程构造一些非退化测度的最优类。