Mathematical Sciences: Problems in Linear and Nonlinear Partial Differential Equations

点击翻译按钮获取中文摘要

Abstract Catlin Catlin will investigate two problems. For the first problem assume that P is a smooth real differential operator of order two such that at each point the principal symbol is semi-definite, although the sign of the symbol is allowed to change from point to point. One can say that P is of finite type if a related system of first order operators is elliptic. Under these hypotheses, Catlin wants to prove that P is hypoelliptic. For the second problem, assume that M is a smooth strictly pseudoconvex CR manifold of dimension 2n-1 with n greater than 3. Then M can be embedded in n-dimensional complex Euclidean space. Catlin will study a family of norms in which the estimates for the Cauchy-Riemann equations are "elliptic" in order to prove a "sharp" embedding theorem. Many physical phenomena such as vibrations or heat flow in an object can be expressed in terms of solutions of partial differential equations. The behavior of these solutions can often be deduced by examining the coefficients of the equations. This proposal is devoted to a study of how the solution changes when certain coefficients change from positive to negative, and also whether the coefficients are irregular. Such equations may be relevant for physical processes in which either the nature of the problem is qualitatively different as one passes from one region to another, or in which the object being modeled has an irregular shape.