Partial Differential Equations; Fundamental Solutions and Applications

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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.