Partial Differential Equations; Fundamental Solutions and Applications

耶鲁大学数学系Richard Beals研究计划“偏微分方程”摘要本项目的主要目标是寻找各种类型的二阶模型线性算子的尽可能显式的解算子：亚椭圆算子，如与复杂域的边界问题相关的算子；退化椭圆算子及相关的退化抛物算子和弱双曲算子；以及混合类型的算子，比如统计物理中出现的算子。这是B.Gaveau和P.C.Greiner的早期工作的延续，并在很大程度上扩展了他们的工作，他们已经为与强和弱伪凸域相关的许多模型亚椭圆算子提供了显式公式和最优参数。在另一个方向上，计划与D.H.Sattinger一起研究完全可积方程，重点关注能量依赖势的谱变换和逆谱变换。大多数物理和几何现象都是用偏微分方程来描述的，通常是用二阶方程。例子包括描述声、光、辐射、扩散、核过程等行为的方程。经典物理模型方程的精确解在数学物理和偏微分方程一般理论的发展中起着核心作用。后一种理论在本世纪取得的进步，很大程度上是由于在理论上和数值上近似更复杂方程的解的技术上的改进。然而，在这些更复杂的方程中，有可能找到精确解的模型，希望精确解将再次在我们更详细的理解中发挥作用。这个建议描述了一些有趣的线性方程，在最近的工作的基础上，有可能得到相当明确的精确解。它还提出了一项广泛研究的技术的进一步工作，该技术提供某些类型的非线性偏微分方程的显式解。在这些方程中，有在某些情况下描述水波和脉冲在光纤中的传播的方程。

DMS-9800605 Abstract of research proposal "Partial Differential Equations" by Richard Beals, Department of Mathematics, Yale University The principal goal of this project is to find solution operators in as explicit a form as possible for model second order linear operators of various types: subelliptic operators, such as those associated to boundary problems in complex domains; degenerate elliptic, and associated degenerate parabolic and weakly hyperbolic operators; and operators of mixed type, such as those that occur in statistical physics. This continues, and expands considerably on, earlier work with B.Gaveau and P.C.Greiner which has produced explicit formulae and optimal parametrices for a number of model subelliptic operators associated to strongly and weakly pseudoconvex domains. In another direction, work is planned with D.H.Sattinger on completely integrable equations, focussing on the spectral and inverse spectral transforms for energy dependent potentials. Most physical and geometric phenomena are described by partial differential equations, typically by equations of second order. Examples include the equations that describe the behavior of sound, light, radiation, diffusion, nuclear processes, and so on. Exact solutions for the model equations of classical physics have played a central role in the development both of mathematical physics and of the general theory of partial differential equations. Much of the advance in the latter theory in this century has been due to refinements in the technique for approximating the solutions of more complex equations, both theoretically and numerically. Nevertheless among these more complex equations are models for which it is possible to find exact solutions, and it is hoped that the exact solutions will again play a role in our understanding at a more detailed level. This proposal describes certain linear equations of interest for which it may be possible, building on recent wor k, to obtain fairly explicit exact solutions. It also proposes further work on a widely studied technique that provides explicit solutions of certain types of nonlinear partial differential equations. Among these equations are those that describe water waves, in certain circumstances, and the propagation of pulses in optical fibers.