Mathematical Sciences: Problems in Linear and Nonlinear Partial Differential Equations

Catlin将研究两个问题。对于第一个问题，假设P是一个光滑的二阶实微分算子，使得在每一点上主符号是半确定的，尽管符号的符号可以在点与点之间改变。如果一阶算子的相关系统是椭圆的，则可以说P是有限型的。在这些假设下，Catlin想要证明P是半椭圆的。对于第二个问题，假设M是一个维数为2n-1且n大于3的光滑严格伪凸CR流形。然后M可以嵌入到n维复欧几里德空间中。卡特林将研究柯西-黎曼方程估计为“椭圆”的一组范数，以证明一个“尖锐的”嵌入定理。许多物理现象，如物体中的振动或热流，都可以用偏微分方程的解来表示。这些解的性质通常可以通过检查方程的系数来推导出来。本提案致力于研究当某些系数由正变为负时，解如何变化，以及系数是否不规则。这些方程可能与物理过程相关，其中问题的性质在从一个区域到另一个区域的过程中是不同的，或者被建模的对象具有不规则的形状。

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.