Optimal Regularity for Nonlinear Pde's and Systems in Carnot-Caratheodory Spaces and Applications to Geometry, Symmetry for Pde's, Unique Continuation

卡诺-卡拉特奥多里空间中非线性偏微分方程和系统的最优正则性及其几何应用、偏微分方程的对称性、唯一延拓

• 批准号: 9706892
• 负责人: Nicola Garofalo
• 金额: $ 11.01万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706892&HistoricalAwards=false
• 关键词: Optimal; Regularity; Nonlinear; Pde; Systems

9706892 Garofalo The proposal is concerned with three projects: 1) Optimal regularity for nonlinear pde's and systems in Carnot-Caratheodory (CC) spaces and the applications of this theory to geometry, isoperimetric and Sobolev inequalities, minimal surfaces in (CC) spaces and their regularity, Dirichlet problem in nilpotent Lie groups: Connection between harmonic measure and perimeter, harmonic maps between CR manifolds. 2) Symmetry in overdetermined boundary value problems, both interior and exterior, in Euclidean and CR geometry, minimizers in the isoperimetric and Sobolev inequalities in the Heisenberg group, a conjecture of E. De Giorgi and its parabolic counterpart. 3) Unique continuation for sub-elliptic operators, applications to scattering, inverse problems on the Heisenberg group, unique continuation for nonlinear equations of p-Laplacian type. The proposed research sits at the confluence of two main areas of interest in mathematics known as partial differential equations and geometry. Both areas find their origin and motivation in problems arising in the observation and description of natural phenomena, at every scale. A main focus of the proposed research is, e.g., the reconstruction of the shape of a body knowing some quantities that can be measured on the surface that surrounds the body. Such a problem has a great relevance in the applied sciences and is especially important in areas of Federal strategic interest ranging from computerized tomography, to biotechnology, to control of the core of a nuclear reactor, etc. The proposed research will also contribute to the development of human resources through the involvement of young investigators (doctoral students). The principal investigator is also writing a book which will focus on those aspects of the program that has been developed over the past few years in collaboration with several doctoral students.

9706892 Garofalo 该提案涉及三个项目： 1) 卡诺-卡拉特奥多里 (CC) 空间中非线性偏微分方程和系统的最优正则性以及该理论在几何、等周和索博列夫不等式、(CC) 空间中的最小曲面及其应用正则性、幂零李群中的狄利克雷问题：调和测度与周长之间的连接、CR 流形之间的调和映射。 2) 欧几里得几何和 CR 几何中超定边值问题（内部和外部）的对称性，海森堡群中等周和索博列夫不等式的最小化，E. De Giorgi 的猜想及其抛物线对应物。 3) 亚椭圆算子的独特延拓，散射的应用，海森堡群的反演问题，p-拉普拉斯型非线性方程的独特延拓。 拟议的研究位于数学中两个主要兴趣领域（即偏微分方程和几何）的交汇处。这两个领域的起源和动机都在于对各种规模的自然现象的观察和描述中出现的问题。所提出的研究的一个主要焦点是，例如，在知道可以在身体周围的表面上测量的一些量的情况下重建身体的形状。这样的问题在应用科学中具有很大的相关性，并且在联邦战略利益领域尤其重要，从计算机断层扫描到生物技术，再到核反应堆核心的控制等。拟议的研究也将有助于发展通过年轻研究者（博士生）的参与来优化人力资源。首席研究员还在写一本书，重点关注过去几年与几位博士生合作开发的项目的各个方面。