Nonlinear Partial Differential Equations in Sub-Riemannian Geometry

亚黎曼几何中的非线性偏微分方程

• 批准号: 0701001
• 负责人: Nicola Garofalo
• 金额: $ 25.49万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701001&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations; Sub

Nonlinear Partial Differential Equations in sub-Riemannian Geometry Abstract of Proposed Research Nicola Garofalo Sub-Riemannian spaces model media with a constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem at hand. Typical examples are crystalline structures, or the movements of the arm of a robot. Models of sub-Riemannian spaces appear in diverse areas of both pure and applied sciences. These include harmonic analysis, several complex variables, group representation, calculus of variations and control theory, geometry (collapsing of Riemannian spaces, CR geometry), geometric measure theory, (Alexandrov spaces, Lie group theory, complex manifolds), quantum mechanics, robotics, mathematical finance, material sciences (crystalline structures), medicine (neurophysiology of the cerebral cortex). The development of analysis and geometry during the past century has been greatly influenced by questions arising from the analysis of systems of partial differential equations; often nonlinear systems. While most of these problems have by now been settled in the classical Euclidean or Riemannian settings, their sub-Riemannian counterparts presently form a body of fundamental and challenging new directions in mathematics. The appropriate mathematical formulation of the problems at hand involves the framework of sub-Riemannian spaces, whose basic prototype is the Heisenberg group (also known to physicists as Weyl group). The class of Carnot groups, is the geometric framework for most problems to be studied under this award. Specific topics include (i) To continue the study of minimal surfaces with particular emphasis on the Bernstein problem and on the question of their regularity including Poincare-Sobolev inequalities and Liouville type theorems on minimal surfaces; (ii) The development of a regularity theory for new variational inequalities with non-holonomic constraints arising in various branches of the applied sciences and the investigation of monotonicity formulas for the relevant constrained energies associated with these problems; (iii) Isoperimetric inequalities for the Gaussian measures associated with the heat semigroup; (iv) To continue the development of the theory of convexity in connection with a maximum principle of Alexandrov-Bakelman-Pucci type; (v) To investigate a CR positive mass theorem using the theory of minimal surfaces; (vi) To study the sharp interior regularity of solutions of nonlinear subelliptic equations. The basic laws that describe most natural phenomena are usually stated as partial differential equations or systems of equations. An understanding of the physical world also requires use of the underlying geometric structure. The present proposal belongs to the mainstream of research which sits at the confluence of the theory of partial differential equations with an emerging type of geometry, called sub-Riemannian geometry. Both theories have witnessed an explosion of interest in the last decade and now attract the interest of various schools of mathematicians both in the US and abroad. Under this award we will investigate problems from mathematical physics and geometry where symmetry plays an important role. Symmetry is present everywhere in nature, including in the fundamental laws of gravitation and electrostatic attraction. The study of conditions under which a natural phenomenon develops symmetries has both practical consequences and intrinsic interest.

亚黎曼几何中的非线性偏微分方程 拟议研究摘要 Nicola Garofalo 亚黎曼空间模型具有约束动力学的介质：任何点仅允许沿着一组有限的方向运动，这些方向由当前的物理问题规定。典型的例子是晶体结构或机器人手臂的运动。亚黎曼空间模型出现在纯科学和应用科学的不同领域。其中包括调和分析、复数变量、群表示、变分和控制理论、几何学（黎曼空间的塌缩、CR几何）、几何测度论、（亚历山德罗夫空间、李群理论、复流形）、量子力学、机器人学、数学金融学、材料科学（晶体结构）、医学（大脑的神经生理学） 皮层）。上个世纪分析和几何的发展受到偏微分方程组分析所产生的问题的极大影响。通常是非线性系统。虽然这些问题中的大多数现在已经在经典欧几里得或黎曼环境中得到解决，但它们的亚黎曼对应问题目前形成了一系列基本且具有挑战性的数学新方向。手头问题的适当数学表述涉及亚黎曼空间的框架，其基本原型是海森堡群（物理学家也称为韦尔群）。卡诺群类是该奖项研究的大多数问题的几何框架。具体主题包括 (i) 继续研究最小曲面，特别强调伯恩斯坦问题及其规律性问题，包括最小曲面上的庞加莱-索博列夫不等式和刘维尔型定理； (ii) 发展应用科学各个分支中出现的具有非完整约束的新变分不等式的正则理论，并研究与这些问题相关的相关约束能量的单调性公式； (iii) 与热半群相关的高斯测度的等周不等式； (iv) 继续发展与 Alexandrov-Bakelman-Pucci 型极大值原理相关的凸性理论； (v) 使用最小曲面理论研究 CR 正质量定理； (vi) 研究非线性亚椭圆方程解的尖锐内部正则性。描述大多数自然现象的基本定律通常被表述为偏微分方程或方程组。对物理世界的理解还需要使用底层的几何结构。目前的建议属于研究的主流，它位于偏微分方程理论与新兴几何类型（称为亚黎曼几何）的融合处。这两种理论在过去十年中引起了人们的极大兴趣，现在吸引了美国和国外各数学家的兴趣。在这个奖项下，我们将研究数学物理和几何中对称性起着重要作用的问题。对称性在自然界中无处不在，包括万有引力和静电引力的基本定律。对自然现象产生对称性的条件的研究既具有实际意义，又具有内在的意义。