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.

次黎曼几何中的非线性偏微分方程组建议研究的摘要尼古拉·加洛法洛次黎曼空间模型介质具有约束动力学：任何点上的运动只允许沿着有限的方向，这是由手头的物理问题所规定的。典型的例子是晶体结构，或机器人手臂的运动。亚黎曼空间的模型出现在理论科学和应用科学的不同领域。这些课程包括调和分析、多个复变量、群表示、变分与控制论、几何(黎曼空间的塌缩，CR几何)、几何测度论(亚历山大空间、李群论、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(大脑皮层的神经生理学)。在过去的一个世纪里，分析和几何的发展很大程度上受到了偏微分方程组(通常是非线性系统)分析中产生的问题的影响。虽然到目前为止，这些问题中的大多数已经在经典的欧几里得或黎曼背景下得到解决，但它们的次黎曼背景下的对应问题目前在数学中形成了一系列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Weyl群)。卡诺群的类，是这个奖项下要研究的大多数问题的几何框架。具体的主题包括：(I)继续研究极小曲面，特别侧重于Bernstein问题及其正则性问题，包括Poincare-Sobolv不等式和极小曲面上的Liouvle型定理；(Ii)发展应用科学各分支中出现的具有非完整约束的新变分不等式的正则性理论，并研究与这些问题相关的约束能量的单调性公式；(Iii)与热半群有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式或方程组。对物理世界的理解也需要使用潜在的几何结构。目前的建议属于偏微分方程组理论与一种新兴的几何类型--亚黎曼几何--交汇处的主流研究。在过去的十年里，这两种理论都引起了人们的极大兴趣，现在吸引了国内外不同学派的数学家的兴趣。在这个奖项下，我们将研究对称性起重要作用的数学、物理和几何问题。对称性在自然界中无处不在，包括引力和静电吸引的基本定律。研究自然现象形成对称性的条件既有实际的结果，也有内在的利益。