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.

拟议研究的次摩曼尼亚次级几何形状摘要中的非线性部分微分方程摘要Nicola Garofalo次 - 里曼尼亚空间模型具有限制动力学的模型介质：在任何时候只允许沿着有限的方向进行运动，而这些方向是由手头的物理问题规定的。典型的例子是晶体结构或机器人手臂的运动。亚里曼尼亚次空间的模型出现在纯科学和应用科学领域。其中包括谐波分析，几个复杂变量，群体表示，变异和控制理论的计算，几何形状（riemannian空间崩溃，CR几何形状），几何措施理论，（Alexandrov空间，Lie compaces，Lie Cloass Theory，复杂的歧管），量子力学，机器人力学，机器人机械师，数学金融，材料科学（晶体结构），医学（大脑皮层的神经生理学）。过去一个世纪的分析和几何形状的发展受到偏微分方程系统分析引起的问题的极大影响。通常是非线性系统。尽管这些问题中的大多数已经解决了古典欧几里得或里曼尼亚的环境，但他们的次曼尼亚人目前构成了数学中基本和挑战性的新方向。手头问题的适当数学表述涉及亚riemannian空间的框架，其基本原型是海森伯格组（也被物理学家称为Weyl组）。 Carnot群体类别是根据该奖项研究的大多数问题的几何框架。具体主题包括（i）继续研究最小表面，特别着重于伯恩斯坦问题以及其规律性问题，包括庞贝罗 - 索伯夫（Poincare-Sobolev）不平等和liouville type定理，以最小的表面进行定理； （ii）开发针对应用科学各个分支的非全面约束的新变异不平等的规则性理论，并研究了与这些问题相关的相关约束能量的单调性公式的研究； （iii）与热半群相关的高斯措施的等距不平等； （iv）继续发展与Alexandrov-Bakelman-Pucci类型的最大原理有关的凸性理论； （v）使用最小表面理论研究Cr阳性质量定理； （vi）研究非线性下层次方程溶液的急剧内部规律性。描述大多数自然现象的基本定律通常被称为部分微分方程或方程系统。对物理世界的理解还需要使用潜在的几何结构。本提案属于研究的主流，该研究位于部分微分方程与新兴几何类型的几何形状（称为亚riemannian几何形状）的汇合处。在过去的十年中，这两种理论都见证了兴趣的爆炸，现在吸引了美国和国外的数学学院各个流派的兴趣。在此奖项下，我们将研究对称性起重要作用的数学物理和几何形状的问题。对称性在自然界中无处不在，包括重力和静电吸引力的基本定律。对自然现象形成对称性的条件的研究既具有实际后果，也具有内在的兴趣。