This paper is concerned with Sturm–Liouville problems (SLPs) with distribution weights and sets up the min–max principle and Lyapunov-type inequality for such problems. As an application, the paper solves the following optimization problems: If the first eigenvalue of a string vibration problem is known, what is the minimal total mass and by which distribution of weight is it attained; if both the first eigenvalue and the total mass are known, what is the corresponding results on the string mass? The vibration problem leads to a SLP with the spectral parameter in both the equation and the boundary conditions. Our main method is to incorporate this problem into the framework of classical SLPs with weights in an appropriate space by transforming it into the one with distribution weight, which provides a different idea for the investigation of the SLPs with spectral parameter in boundary condition.

中文翻译：

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涉及分布权重的 Sturm-Liouville 问题及其在优化问题中的应用

本文关注具有分布权重的 Sturm-Liouville 问题 (SLP)，并针对此类问题建立了最小-最大原理和李雅普诺夫型不等式。作为一个应用，本文解决了以下优化问题： 如果弦振动问题的第一个特征值已知，最小总质量是多少，重量分布是什么；如果第一个特征值和总质量都已知，那么弦质量的相应结果是什么？振动问题导致在方程和边界条件中都具有谱参数的 SLP。我们的主要方法是通过将这个问题转换为具有分布权重的问题，将这个问题纳入在适当空间中具有权重的经典 SLP 框架中，