Eigenvalues for Vibrating Plates and for Thin Film Equations

DMS-9970228ABSTRACT本研究试图建立以下三个猜想。(A)在给定固定面积的情况下，受横向拉力的振动固支板的基音对于圆板而言是最小的。在数学上，这涉及到在施加Dirichlet和Neumann边界条件的情况下，最小化给定区域的平面域之间的biLaplace算子的第一本征值。将采用对称化方法。(B)“薄膜型”偏微分方程组的稳态在某些情况下可以是线性稳定的，但在其他情况下可能是不稳定的，并倾向于膜夹断或破裂。将使用试函数和微分不等式技术来证明不稳定结果。(C)对于直线段，平衡单位电荷分布在平面上关于其静电质心的一阶矩是最大的。将开发一个“双*函数”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、流体和电荷问题的研究已有数百年的历史。今天，它们仍然具有挑战性和重要性，因为科学家们正在极端规模和极端条件下努力处理新旧材料。然而，20世纪科学史表明，在实际问题上的进展往往依赖于基础研究中发展起来的方法和见解。选择本提案中涉及的问题是因为它们具有提供此类见解的潜力。例如，对问题(A)和(C)的猜测答案是“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增强我们对振动和电荷分布问题的理解和能力。更具体地说，方案中的第一个问题涉及选择振动刚性薄板的形状，以使最低共振频率最小化。100多年前的实验表明，圆盘是频率最低的，但仍然没有理论上的解释。下一个问题涉及对粘性油的薄层润滑膜进行建模的方程。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。非线性问题是数学分析的前沿，本研究旨在了解非线性效应是如何在薄膜流体方程中产生稳定性、不稳定性和夹点的。提案中考虑的第三个问题是一个由静电学理论产生的简单但仍未解决的问题，即电子如何在相互排斥的情况下在导体上排列。

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.