Partial Differential Equations; Fundamental Solutions and Applications

DMS-9800605研究提案摘要“偏微分方程” 作者：Richard Beals，耶鲁大学数学系 这个项目的主要目标是找到解决方案运营商 以尽可能明确形式来模拟二阶线性 各种类型的算子：次椭圆算子，例如与复域中的边界问题相关的算子;退化椭圆算子， 和相关的退化抛物和弱双曲算子;和混合型算子，如那些发生在统计 物理学 这是对早期工作的继续和扩展 与B.Gaveau和P.C.Greiner合作， 一类模型次椭圆算子的最优参数 强伪凸域和弱伪凸域。 在另一个方向上，计划与D.H.Sattinger一起研究完全可积的 方程，侧重于频谱和逆频谱变换 能量依赖的潜力。 大多数物理和几何现象都是用局部的 微分方程，通常是二阶方程。 示例包括描述以下行为的方程： 声、光、辐射、扩散、核过程等等。 经典物理模型方程的精确解有 在数学物理和偏微分方程的一般理论的发展中发挥了核心作用。 后一种理论在本世纪的许多进展是由于 在近似的解决方案的技术的改进 更复杂的方程，无论是理论上还是数值上。 然而，在这些更复杂的方程中， 有可能找到确切的解决办法，希望 精确解将再次在我们的理解中发挥作用， 详细水平。 这个建议描述了某些线性方程， 在最近工作的基础上， 得到相当明确的精确解。 它还建议进一步 致力于一项广泛研究的技术，该技术提供了明确的解决方案 某些类型的非线性偏微分方程。 之间 这些方程是描述水波的方程，在某些情况下， 在光纤中的脉冲的传播。

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.