Aspects of Subelliptic Partial Differential Equations

次椭圆偏微分方程的各个方面

• 批准号: 9623007
• 负责人: Francis Christ
• 金额: $ 19.36万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623007&HistoricalAwards=false
• 关键词: Aspects; Subelliptic; Partial; Differential; Equations

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.

抽象基督 研究将进行各种问题有关的次椭圆型偏微分方程和相关主题。其中包括（i）满足括号条件的向量场平方和的局部和全局解析亚椭圆性，以及d-杆Neumann问题。(ii)d-杆Neumann问题和Bergman投影在无穷可微（但不是有限型）范畴中的整体正则性。(iii)向量场平方和Gevrey正则性最优指数的刻画。 (iv)具有双特征和光滑系数的偏微分算子的局部可解性。 所有这些主题都与某些特征值问题有关，这些问题是由偏微分算子的极限和/或约化引起的;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了原始特征值。 运营商重点将放在低维问题，以及那些连接到复杂的分析，在几个变量和调和分析李群。 自从艾萨克·牛顿洞察到自然的基本定律应该与观察到的量变化的速率有关，而不是直接描述这些量，大多数基本物理定律都被表述为偏微分方程。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。除了在非常对称的情况下，偏微分方程很少能用显式公式求解。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），以及如果有的话，解可以表现出什么样的奇点（冲击波，漩涡和波前是数学奇点的典型物理表现）。 这些问题通常可以在无法构建解决方案的公式的情况下理解。然而，偏微分算子在性质上变化很大，每种类型都必须根据自己的条件进行研究。