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

这一建议涉及非线性偏微分方程和几何交界处的一些问题，特别强调次黎曼流形。统一的主题是系统地寻找手头问题解的一些基本单调性性质。这样的性质在分析和几何中发挥着特殊的作用，并经常导致对相关方程的性质的显著洞察。这个建议的主要方向之一是在次黎曼几何中提出一个新的曲率概念。它是黎曼几何中Ricci曲率张量的推广。将新的Bochner恒等式与某些类熵泛函的单调性相结合，对于广义Ricci张量为非负的流形，在严格正的情形下得到了Li-Yau型先验梯度界、Harnack不等式、Gauss型上界、等周不等式和次黎曼Bonnet-Myers紧性定理.在另一个方向上，该建议旨在进一步加深次黎曼几何中极小曲面的现有知识，特别强调次黎曼伯恩斯坦问题。PI和他的合著者最近在第一个(三维)海森堡群中解决了这个问题。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的单调性的研究。在另一个方向上，研究了障碍被限制在低维流形上的椭圆型和抛物型变分不等式解的一些新的单调性。这样的单调性公式被应用于研究相关自由边界问题的正则性。这一建议可以放在数学的两个主要研究领域--偏微分方程式和黎曼几何的交汇处。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中可观察到的现象。黎曼几何提供了一个框架，它是理解当我们面对牛顿和伽利莱经典力学之外的现象时所发生的事情所必需的。例如，在爱因斯坦-S的相对论中，对弯曲时空的描述需要使用黎曼流形及其内在几何。在过去的十年里，人们对黎曼几何的深远推广以及描述这一领域出现的新现象所需的相关偏微分方程式的兴趣激增。由于这一提议处于其中一些发展的前沿，它有可能影响作为这些进步的起源地的数学和应用科学领域(机器人、机械工程、神经科学)。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、出版物和网站系统地传播相关研究，本提案是人力资源开发的重要组成部分。

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.