Mathematical Sciences: Isoperimetric Inequalities for Eigenvalues in Mathematical Physics and Geometry

Ashbaugh DMS-9500968 阿什堡将进行的研究涉及寻找最佳界限的特征值的经典偏微分算子产生的数学物理和微分几何。 具体的例子包括瑞利猜想的最低本征值的固支板的所有维度，和佩恩-波利亚-温伯格猜想的比例特征值的Dirichlet拉普拉斯算子的任意有界域在欧几里德n-空间。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。

Ashbaugh DMS-9500968 The research that Ashbaugh will conduct concerns finding optimal bounds for the eigenvalues of the classical partial differential operators arising in mathematical physics and differential geometry. Specific examples include Rayleigh's conjecture for the lowest eigenvalue of the clamped plate for all dimensions, and the Payne-Polya-Weinberger conjecture on the ratio of eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain in Euclidean n-space. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.