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学的一些问题，特别强调了次黎曼流形。统一的主题是一些基本的单调性质的解决方案的手的系统搜索。这些性质在分析和几何学中起着特殊的作用，并且经常导致对相关方程性质的显著洞察。其中一个主要的方向，在这个建议是一个新的概念，曲率次黎曼几何。它代表了黎曼几何的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，对于这种广义Ricci张量为非负的流形，得到了Li-Yau型先验梯度界、Harnack不等式、Gaussian上界、等周不等式以及严格正情形下的次黎曼Bonnet-Myers紧性定理.在另一个方向上，该建议旨在进一步促进 目前的知识极小曲面在分黎曼几何与特别强调分黎曼伯恩斯坦问题。PI和他的合著者最近在第一（三维）海森堡群中解决了这个问题。所提出的研究围绕着高维问题的分析，以及相关的面积泛函的新的单调性性质的研究。在另一个方向的建议是关注的研究一些新的单调性的椭圆和抛物型变分不等式的解决方案的障碍局限于位于低维流形。这种单调性公式然后被应用到相关的自由边界问题的正则性的研究中。这个建议可以被放置在两个主要的数学研究领域的交汇处，即偏微分方程和黎曼几何。偏微分方程是一个未知函数和它的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，这是理解当我们面对牛顿和伽利略经典力学之外的现象时所必需的。例如，在爱因斯坦？在相对论中，弯曲时空的描述需要使用黎曼流形及其内在几何。在过去的十年里，人们对黎曼几何的广泛推广以及相关的偏微分方程的兴趣激增，这些方程是描述这一领域出现的新现象所必需的。由于这一建议是在这些发展的最前沿，它有可能影响这些领域的数学和应用科学（机器人，机械工程，神经科学），这些领域是这些进步的起源。鉴于博士生和博士后的广泛参与和培训，以及通过研讨会、讲座、会议、出版物和网站系统地传播有关研究，这项建议是人力资源开发的一个重要组成部分。

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.