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 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周长，Cr歧管之间的谐波图。 2）在欧几里得和CR几何形状中，内部和外部的超确定边界价值问题中的对称性，在海森伯格集团的等值仪和索波列夫的不平等中，在E. de Giorgi及其寄生虫对应物中的猜想。 3）对于亚椭圆算子的独特延续，在海森堡组上的散射，逆问题的应用，p-laplacian类型的非线性方程的独特延续。 拟议的研究是两个主要的数学感兴趣领域的融合，称为部分微分方程和几何形状。这两个领域都在每个规模的自然现象的观察和描述中都发现了它们的起源和动力。拟议的研究的一个主要重点是，例如，人体形状的重建，知道可以在周围身体的表面上测量的一些数量。这样的问题在应用科学方面具有很大的意义，在从计算机层析成像到生物技术到控制核反应堆核心等领域的联邦战略兴趣方面尤为重要。拟议的研究还将通过年轻研究人员的参与（博士生）来为人力资源的发展做出贡献。首席调查员还在撰写一本书，该书将重点关注过去几年与几个博士生合作开发的计划的那些方面。