Monotonicity formulas, nonlinear PDE's and sub-Riemannian Geometry

单调性公式、非线性偏微分方程和亚黎曼几何

• 批准号: 1001317
• 负责人: Nicola Garofalo
• 金额: $ 30万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2013-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001317&HistoricalAwards=false
• 关键词: Monotonicity; formulas; nonlinear; PDE; sub

This proposal is concerned with a number of questions at the interface of nonlinear partial differential equations and geometry, with particular emphasis on sub-Riemannian manifolds. The unifying theme is the systematic search of some basic monotonicity properties of the solutions of the problem at hand. Such properties play a special role in analysis and geometry and often lead to a remarkable insight in the nature of the relevant equations. One of the main directions in this proposal is a new notion of curvature in sub-Riemannian geometry. It represents a generalization of the Ricci curvature tensor from Riemannian geometry. Combining new Bochner identities with the monotonicity of some entropy-like functionals, for manifolds for which such generalized Ricci tensor is nonnegative one is led to a priori gradient bounds of Li-Yau type, Harnack inequalities, Gaussian upper bounds, isoperimetric inequalities, and a sub-Riemannian Bonnet-Myers compactness theorem in the strictly positive case. In another direction the proposal aims at furthering the present knowledge of minimal surfaces in sub-Riemannian geometry with particular emphasis on the sub-Riemannian Bernstein problem. The PI and his co-authors have recently solved this problem in the first (three-dimensional) Heisenberg group. The proposed research revolves around the analysis of the higher dimensional problem as well as the study of new monotonicity properties of the relevant area functionals. In yet another direction the proposal is concerned with the study of some new monotonicity properties of solutions of variational inequalities of elliptic and parabolic type with an obstacle confined to lie in alower dimensional manifold. Such monotonicity formulas are then applied to the study of the regularity of the relevant free boundary problems.This proposal can be placed at the confluence of two major areas of research in mathematics known as partial differential equations and Riemannian geometry. Partial differential equations are relations between an unknown function and a certain number of its derivatives. They govern the observable phenomena of the physical world. Riemannian geometry provides with a framework which is necessary to understand what happens when we are confronted with phenomena which fall outside the classical mechanics of Newton and Galilei. For instance, in Einstein?s theory of relativity the description of the curved space-time requires the use of Riemannian manifolds, with their intrinsic geometry. The past decade has witnessed an explosion of interest in a far reaching generalization of Riemannian geometry, as well as in the relevant partial differential equations which are needed to describe the new phenomena which arise in this area. Since this proposal is at the forefront of some of these developments it has the potential to impact those areas of mathematics and of the applied sciences (robotics, mechanical engineering, neuroscience) which are at the origin of these advances. In view of the extensive involvement and training of doctoral students and post-doctoral advisee, and the systematic dissemination of the relevant research through seminars, lectures, conferences, publications and websites, this proposal presents a strong component of human resources development.

该提案涉及非线性偏微分方程和几何接口的许多问题，特别强调亚黎曼流形。统一的主题是系统地搜索当前问题的解决方案的一些基本单调性属性。这些性质在分析和几何中起着特殊的作用，并且通常会导致对相关方程本质的深刻洞察。该提案的主要方向之一是亚黎曼几何中曲率的新概念。它代表了黎曼几何中的里奇曲率张量的推广。将新的博赫纳恒等式与一些类熵泛函的单调性相结合，对于此类广义里奇张量为非负的流形，可得出 Li-Yau 型的先验梯度界限、Harnack 不等式、高斯上限、等周不等式和严格正情况下的亚黎曼 Bonnet-Myers 紧性定理。在另一个方向上，该提案旨在进一步加深对亚黎曼几何中最小曲面的了解，特别强调亚黎曼伯恩斯坦问题。 PI 和他的合著者最近在第一（三维）海森堡小组中解决了这个问题。所提出的研究围绕高维问题的分析以及相关面积泛函的新单调性性质的研究。在另一个方向上，该提案涉及研究椭圆型和抛物线型变分不等式解的一些新的单调性性质，其中障碍仅限于较低维流形。然后将这种单调性公式应用于相关自由边界问题的规律性研究。该提议可以放在偏微分方程和黎曼几何这两个主要数学研究领域的交汇处。偏微分方程是未知函数与其一定数量的导数之间的关系。它们支配着物理世界的可观察现象。黎曼几何提供了一个框架，对于理解当我们遇到牛顿和伽利略经典力学之外的现象时会发生什么是必要的。例如，在爱因斯坦的相对论中，弯曲时空的描述需要使用黎曼流形及其内在几何结构。在过去的十年里，人们对黎曼几何的深远推广以及描述该领域出现的新现象所需的相关偏微分方程的兴趣激增。由于该提案处于其中一些发展的前沿，因此它有可能影响作为这些进步起源的数学和应用科学（机器人、机械工程、神经科学）领域。鉴于博士生和博士后导师的广泛参与和培训，以及通过研讨会、讲座、会议、出版物和网站系统传播相关研究成果，该提案成为人力资源开发的重要组成部分。