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.

非线性偏微分方程在次黎曼几何中的抽象研究 尼古拉·加罗法洛（Nicola Garofalo）亚Riemannian空间对具有约束动力学的介质进行建模：任何点的运动都只允许沿着有限的一组方向，这些方向由手头的物理问题规定。典型的例子是晶体结构，或机器人手臂的运动。次黎曼空间的模型出现在纯科学和应用科学的不同领域。这些包括调和分析，几个复杂的变量，群表示，变分法和控制理论，几何（黎曼空间的崩溃，CR几何），几何测量理论，（亚历山德罗夫空间，李群理论，复杂的流形），量子力学，机器人，数学金融，材料科学（晶体结构），医学（大脑皮层的神经生理学）。在过去的世纪中，分析和几何的发展受到了偏微分方程系统（通常是非线性系统）分析问题的极大影响。虽然这些问题中的大多数现在已经在经典的欧几里得或黎曼环境中得到解决，但它们的次黎曼对应物目前形成了数学中基本的和具有挑战性的新方向。手头问题的适当数学表述涉及次黎曼空间的框架，其基本原型是海森堡群（物理学家也称为外尔群）。卡诺群的类，是大多数问题的几何框架下研究这个奖项。具体议题包括（i）继续研究极小曲面，特别强调伯恩斯坦问题及其正则性问题，包括Poincare-Sobolev不等式和关于极小曲面的Liouville型定理;（ii）发展了一个新的变分不等式的正则性理论，完整约束出现在应用科学的各个分支和调查的单调性公式的有关约束能量与这些问题;（iii）与热半群有关的Gauss测度的等周不等式;（iv）继续发展与Alexandrov-Bakelman-Pucci型极大值原理有关的凸性理论;（v）利用极小曲面理论研究CR正质量定理;（vi）研究非线性次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。理解物理世界也需要使用基本的几何结构。目前的建议属于主流的研究，它坐落在汇合处的理论偏微分方程与新兴类型的几何，所谓的次黎曼几何。这两个理论都见证了爆炸的兴趣在过去的十年里，现在吸引了各种学校的数学家在美国和国外的兴趣。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，包括引力和静电引力的基本定律。研究自然现象发展对称性的条件既有实际意义，也有内在的意义。