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

Ashbaugh DMS-9500968，Ashbaugh将进行的研究涉及为数学物理和微分几何中出现的经典偏微分算子的特征值寻找最优界。具体的例子包括：Rayleigh关于各维固支板最低本征值的猜想，以及关于欧几里得n-空间中任意有界域上Dirichlet Laplacian的特征值之比的Payne-Polya-Weinberger猜想。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

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.