Optimal Bounds for the Characteristic Frequencies of Vibrating Membranes

发展中国家科学奖将支持哥伦比亚密苏里大学的Mark Ashbaugh教授和智利天主教大学的Raphael Benguria教授的研究合作。该项目旨在解决与圆桶的数学尺寸有关的猜想。研究人员建议在“翘曲”鼓中为第一泛音与基本音的比率建立良好的界限，这些鼓在三维空间中跨越一个封闭的曲线。他们还将考虑其他类型的表面，例如球面的一部分，这对曲面的几何非常重要。通过早期合作中使用的技术，研究人员也将检查其他色调比率的界限，以改进现有的界限，并可能获得最佳界限，以及提供最佳结果的形状。该研究有望对微分算子的特征值界的一般领域产生实质性的贡献。美方将从与该领域一位智利专家的持续合作中获益。智利方面将从与美国同行的继续直接参与中受益。

This Science in Developing Countries award will support the research collaboration of Professor Mark Ashbaugh of the university of Missouri, Columbia, and Professor Raphael Benguria of the Pontificia Universidad Catolica de Chile. The project aims to solve conjectures related to the mathematical dimensions of circular drums. The researchers propose to develop good bounds for the ratio of the first overtone to the fundamental tone in "warped" drums, which span a closed curve in three- dimensional space. They will also consider other types of surfaces, such as a piece of a sphere's surface, which are of great importance to the geometry of curved surfaces. Through techniques used in earlier collaborations, the investigators will examine bounds for ratios o other tones as well, in order to improve existing bounds and perhaps obtain the optimal bound, along with the shape that gives the optimal result. The research is expected to yield substantial contributions to the general area of eigenvalue bounds for differential operators. The u.S. side will gain from continued collaboration with a Chilean expert in this area. The Chilean side will benefit from continued direct participation with a U.S. counterpart.