Eigenvalues for Vibrating Plates and for Thin Film Equations

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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.