Eigenvalues for Vibrating Plates and for Thin Film Equations

本研究将试图建立以下三个猜想。(a)给定固定面积的圆板，在侧向张力作用下振动夹紧板的基频最小。在数学上，这涉及在给定面积的平面域中最小化bilaplacian的第一个特征值，同时施加狄利克雷和诺伊曼边界条件。将采用对称方法。(b) “薄流体膜”型偏微分方程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，并且倾向于薄膜夹断或爆炸。将使用试函数和微分不等式技术来证明in/stability结果。(c)平面上平衡单位电荷分布的第一个力矩，关于它的静电质心，对于线段是最大的。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、流体和电荷问题已经研究了数百年。今天，当科学家们在极端尺度和极端条件下努力研究新旧材料时，它们仍然具有挑战性和重要性。然而，20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。之所以选择本建议中讨论的问题，是因为它们有可能提供这种见解。例如，问题(a)和(c)的推测答案是“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，肯定会有利于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，第一个问题是选择一个薄的振动刚性板的形状，以使最低的谐振频率最小。100多年前的实验表明，圆形板是频率最低的一个，但仍然没有理论解释。下一个问题是关于粘性油的薄润滑膜的方程。这些方程是非线性的，这意味着我们熟悉的叠加原理（将两个解相加得到另一个解）并不适用。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流体膜方程中产生稳定、不稳定和夹断的。提案中考虑的第三个问题是由静电理论引起的一个简单但尚未解决的问题，即电子在相互排斥下如何在导体上排列自己。

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.