论一般希尔伯特空间中的波类微分方程：作为工程科学应用的梁的欧拉-伯努利弯曲振动的泛函分析研究
On Wave-Like Differential Equations in General Hilbert Space. The Functional Analytic Investigation of Euler-Bernoulli Bending Vibrations of a Beam as an Application in Engineering Science
AI总结：
本文在一般希尔伯特空间泛函分析框架下研究波类微分方程，重点严格构造并分析了一维弹性梁欧拉-伯努利弯曲振动在不同边界条件下的四阶正自伴微分算子，并与弦的自由波动进行了比较。
摘要：
波类偏微分方程出现在许多工程应用中。本文将工程设置嵌入到现代数学物理泛函分析的希尔伯特空间框架中。“波类”这一概念是对原始波动（偏）微分方程的推广。本文简要概述了物理学和工程学中的三个波类问题，描述了在 $\mathrm{L}^2$-希尔伯特空间中构造正自伴微分算子的数学步骤，这些算子随后可用于波类微分方程，并总结了泛函分析文献中的一些一般性结果。主要部分研究了一维空间中细长、直、弹性梁的自由欧拉-伯努利弯曲振动在 $\mathrm{L}^2$-希尔伯特空间设置下的问题。通过选取适当的索伯列夫空间，我们对相应的四阶（空间）正自伴微分算子进行了数学上严格的引入和分析，这些算子对应于静力学中作为支座出现的不同边界条件。此外，还补充了与弦的自由波摆动（使用拉普拉斯算子作为微分算子）的比较。
英文摘要：
Wave-like partial differential equations occur in many engineering applications. Here the engineering setup is embedded into the Hilbert space framework of functional analysis of modern mathematical physics. The notion wave-like is a generalization of the primary wave (partial) differential equation. A short overview over three wave-like problems in physics and engineering is presented. The mathematical procedure for achieving positive, selfadjoint differential operators in an $\mathrm{L}^2$-Hilbert space is described, operators which then may be taken for wave-like differential equations. Also some general results from the functional analytic literature are summarized. The main part concerns the investigation of the free Euler--Bernoulli bending vibrations of a slender, straight, elastic beam in one spatial dimension in the $\mathrm{L}^2$-Hilbert space setup. Taking suitable Sobolev spaces we perform the mathematically exact introduction and analysis of the corresponding (spatial) positive, selfadjoint differential operators of $4$-th order, which belong to the different boundary conditions arising as supports in statics. A comparison with free wave swinging of a string is added, using a Laplacian as differential operator.
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