Eigenvalues for Vibrating Plates and for Thin Film Equations

振动板和薄膜方程的特征值

• 批准号: 9970228
• 负责人: Richard Laugesen
• 金额: $ 5.88万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970228&HistoricalAwards=false
• 关键词: Eigenvalues; Vibrating; Plates; Thin; Film

DMS-9970228ABSTRACTThe research will attempt to establish the folowing three conjectures. (a) That the fundamental tone of a vibrating clamped plate under lateral tension is minimal for the circular plate, given fixed area. Mathematically, this involves minimizing the first eigenvalue of thebiLaplacian amongst plane domains of given area, with both Dirichlet and Neumann boundary conditions imposed. Symmetrization methods will be employed. (b) That steady states of "thin fluid film" type partial differential equations can be linearly stable in certain situations, but in other situations can be unstable and tend to film pinch-off or blow-up. Trial function and differential inequality techniques will be used to prove the in/stability results. (c) That the first moment of an equilibrium unit charge distribution in the plane, about its electrostatic centroid, is maximal for a line segment. A "double *-function" will be developed to attack this problem.The methods to be applied to these problems are mathematical, though each problem has physical meaning. Problems of vibration, fluids and charges have been studied for hundreds of years. Today they remain challenging and important, as scientists grapple with new and old materials on extreme scales and under extreme conditions. The history of 20th century science shows, though, that progress on practical problems often depends on the methods and insights developed in basic research. The problems addressed in this proposal have been chosen for their potential to provide such insights. For example, the conjectured answers to problems (a) and (c) are "obvious" - the puzzle (and challenge) is that no-one can explain logically why these answers are correct. Finding such a logical explanation is sure to profitably enhance our understanding of and capabilities with problems of vibration and charge distribution. More specifically, the first problem in the proposal deals with choosing the shape of a thin vibrating rigid plate so as to minimize the lowest resonant frequency. Experiments performed over 100 years ago suggest the circular plate is the one with lowest frequency, but there is still no theoretical explanation for this. The next problem concerns the equations that model thin lubricating films of viscous oils. These equations are nonlinear, meaning familiar superposition principles (adding two solutions to get another solution) do not apply. Nonlinear problems are at the forefront of mathematical analysis, and this research aims to understand how it is that nonlinear effects create stability, instability and pinch-off in thin fluid film equations. The third problem considered in the proposal is a simply-stated but still-unsolved problem arising from the theory of electrostatics, to do with how electrons arrange themselves on a conductor under their mutual repulsion.

本研究试图建立以下三种结构。 (a)在给定面积的情况下，受横向拉伸的固支振动板的基频对于圆板是最小的。在数学上，这涉及到最小化的第一特征值的双拉普拉斯之间的平面域的给定面积，与狄利克雷和诺依曼边界条件。将采用对称化方法。 (b)“薄液膜”型偏微分方程的定常状态在某些情况下可以是线性稳定的，但在另一些情况下可能是不稳定的，并趋于膜夹断或爆破。尝试函数和微分不等式技术将被用来证明在/稳定的结果。 (c)平面内平衡单位电荷分布关于其静电质心的一阶矩对线段最大。一个“双 * 函数”将被开发来解决这个问题。应用于这些问题的方法是数学的，尽管每个问题都有物理意义。 振动、流体和电荷问题已经研究了几百年。 今天，它们仍然具有挑战性和重要性，因为科学家们在极端规模和极端条件下与新老材料作斗争。 然而，世纪的科学史表明，实际问题的进展往往取决于基础研究中发展起来的方法和见解。本提案中所涉及的问题是根据其提供此类见解的潜力而选择的。例如，问题（a）和（c）的明确答案是“显而易见的”-难题（和挑战）是没有人能从逻辑上解释为什么这些答案是正确的。找到这样一个合乎逻辑的解释肯定会有益地提高我们对振动和电荷分布问题的理解和能力。更具体地说，第一个问题的建议涉及选择一个薄的振动刚性板的形状，以尽量减少最低的共振频率。100多年前进行的实验表明，圆板是频率最低的板，但仍然没有理论解释。下一个问题是关于粘性油薄润滑膜的模型方程。这些方程是非线性的，这意味着熟悉的叠加原理（将两个解相加得到另一个解）不适用。非线性问题处于数学分析的前沿，本研究旨在了解非线性效应如何在薄流体膜方程中产生稳定性，不稳定性和夹断。该提案中考虑的第三个问题是一个简单但尚未解决的问题，它来自静电理论，即电子如何在相互排斥的情况下在导体上排列。