Aspects of Subelliptic Partial Differential Equations

摘要《基督研究》将对有关次椭圆型偏微分方程式及相关课题的各种问题进行研究。其中包括(I)满足括号条件的向量场平方和的局部和整体解析亚椭圆性，以及d-bar Neumann问题。(Ii)d-bar Neumann问题和Bergman投影在无穷可微(但非有限类型)范畴中的全局正则性。(3)向量场平方和的Gevrey正则性的最优指数刻画。(4)具有双特征和光滑系数的偏微分算子的局部可解性。所有这些主题都与由偏微分算子的极限和/或约化引起的某些特征值问题有关；关键问题是特征值是否存在，以及极限问题在多大程度上控制原始算子。重点将放在低维问题上，以及那些与多变量复变分析和李群上的调和分析有关的问题上。自从艾萨克·牛顿洞察到自然界的基本定律应该与观测量的变化速度相关，而不是直接描述这些量以来，大多数基本的物理定律都被表述为偏微分方程式。因此，理解这类方程及其解已成为数学研究的基本目标之一。偏微分方程很少能用显式公式求解，除非在非常对称的情况下。另一方面，在物理应用中，了解解的某些定性特征通常是有用的：是否存在任何解(这在实践中可能意味着某些情况在物理上是可行的或不可行的)，以及解可以表现出什么样的奇点(激波、漩涡和波前是数学奇点的典型物理表现)。在无法构造解决方案的公式的情况下，通常可以理解这些问题。然而，偏微分算子在性质上有很大的不同，每种类型都必须根据其自己的条件进行研究。

Abstract Christ Research will be conducted on a variety of questions concerning subelliptic partial differential equations and related topics. Among these are (i) Local and global analytic hypoellipticity of sums of squares of vector fields satisfying the bracket condition, and in the d-bar Neumann problem. (ii) Global regularity in the d-bar Neumann problem and for the Bergman projection in the infinitely differentiable (but not finite type) category. (iii) Characterization of optimal exponents of Gevrey regularity for sums of squares of vector fields. (iv) Local solvability of partial differential operators with double characteristics and smooth coefficients. All these topics are related to certain eigenvalue problems that arise from limits and/or reductions of partial differential operators; key issues are whether eigenvalues exist, and the extent to which the limiting problems control the original operators. The emphasis will be on low-dimensional problems, and on those connected to complex analysis in several variables and to harmonic analysis on Lie groups. Since Isaac Newton's insight that the fundamental laws of nature should relate the rates at which observed quantities change, rather than describe those quantities directly, most basic physical laws have been formulated as partial differential equations. To understand such equations and their solutions has thus become one of the fundamental goals of mathematical research. Partial differential equations are rarely solvable by explicit formulas, except in very symmetric situations. On the other hand, in physical applications it is often useful to understand certain qualitative features of solutions: whether any solution exists (which in practice may mean that some situation is or is not physically viable), and what kinds of singularities, if any, solutions can exhibit (shock waves, vortices, and wave fronts are typical physical manifestations of mathematical singularities). These issues can often be understand in situations where formulas for the solutions cannot be constructed. However, partial differential operators vary greatly in character, and each type must be investigated on its own terms.