Some nonlinear problems in analysis and geometry

分析和几何中的一些非线性问题

• 批准号: 0300477
• 负责人: Nicola Garofalo
• 金额: $ 23.8万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300477&HistoricalAwards=false
• 关键词: Some; nonlinear; problems; analysis; geometry

PI: Nicola Garofalo, Purdue UniversityDMS-0300477Abstract:The development of analysis and geometry during the past century has been greatly influenced by the desire of solving various basic problems involving some special partial differential equations, mostly of nonlinear type. 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 open questions. One of the broader objectives of this proposal is to study some of them. This PI is concerned with developing a new theory of minimal surfaces, or more in general surfaces with bounded mean curvature, in sub-Riemannian spaces, study their regularity and classify the isoperimetric sets in some model spaces with symmetries. He proposes a calculus on hypersurfaces which hinges on the idea of horizontal Gauss map, and leads to a new notion of mean curvature The analysis of the ensuing nonlinear equations and systems constitutes a challenging new avenue of study. Within such calculus, minimal surfaces are thus hypersurfaces of zero mean curvature, and a problem of fundamental interest is a sub-Riemannian version of the famous conjecture of Bernstein. The latter displays a marked discrepancy with its classical ancestor and there is a host of new geometric phenomena connected with the singularities of the Gauss map which generically occur at those points where the subbundle which generates the sub-Riemannian structure becomes part of the tangent space to the hypersurface. Given the role of the classical Bernstein problem in the development of last century's mathematics, it is foreseeable that the theory of sub-Riemannian minimal surfaces and the corresponding Bernstein problem will sparkle a broad development. The PI also proposes to find the minimizers in the Folland-Stein embedding for groups of Heisenberg type and Siegel domain of type 2, and thereby compute the best constants. This program is instrumental to attacking the compact CR Yamabe problem for CR manifolds of higher codimension. In connection with the CR Yamabe problem the PI proposes to investigate a CR version of the positive mass theorem from relativity due to Schoen and Yau. It is expected that the theory of minimal surfaces previously mentioned will play an important role. Another emerging theory in sub-Riemannian geometry is that of equations of Monge-Amp\`ere type, which occupy a central position in geometry as well as in the calculus of variations in view of their tight connection with the problem of mass transport. The PI proposes to investigate a new estimate connected with a sub-Riemannian version of the geometric maximum principle of Alexandrov, Bakelman, and Pucci. In joint work he has recently obtained results for the appropriate class of ``convex" functions, and, inspired by N.Krylov's approach, established monotonicity type results for a functional involving the symmetrized horizontal Hessian along with some appropriate commutators. Another problem included in this proposal is the optimal regularity for nonlinear equations arising in the study of quasiregular mappings between Carnot groups. This is presently a fundamental open question and, without its solution, it will be impossible to make substantial advances in nonlinear potential theory for sub-Riemannian spaces. In this connection the PI also plans to analyze the delicate question of the uniqueness of the fundamental solution and Green function, and study the geometric properties of their level sets. Other directions of investigation are the analysis of boundary value problems (Dirichlet, Neumann) for subelliptic equations and their associated heat flows, the study of free boundary problems, and the analysis of global properties of solutions to some pde's arising in geometry and mathematical physics. Partial differential equations and systems formed by the latter are the basic laws, which describe most natural phenomena. An understanding of the physical world also requires grasping the underlying geometric structure of the latter in its various forms. The present proposal belongs to the mainstream of research, which sits at the confluence of the theory of partial differential equations and systems, mostly of nonlinear type, and their connections with an emerging type of geometry, called sub-Riemannian geometry. Both theories have witnessed an explosion of interest in the last decade and they continue to attract the interest of various schools of mathematicians both nationwide and abroad. This proposal is also concerned with problems from mathematical physics and geometry in which symmetry plays an important role. Symmetry is present everywhere in nature, a remarkable instance being the fundamental laws of gravitation and electrostatic attraction. The study of conditions under which a natural phenomenon develops symmetries is important both for practical consequences and for its implications in the furthering of our knowledge.

PI：Nicola Garofalo，普渡大学DMS-0300477摘要：过去一个世纪分析和几何的发展受到解决涉及一些特殊偏微分方程（大多数是非线性类型）的各种基本问题的渴望的极大影响。虽然这些问题中的大多数现在已经在经典欧几里得或黎曼环境中得到解决，但它们的亚黎曼对应问题目前形成了一系列基本的开放问题。该提案更广泛的目标之一是研究其中的一些。该 PI 涉及开发一种新的最小曲面理论，或者更多的是在亚黎曼空间中具有有界平均曲率的一般曲面，研究它们的规律性并对某些具有对称性的模型空间中的等周集进行分类。他提出了一种基于水平高斯图思想的超曲面微积分，并引出了平均曲率的新概念。对随后的非线性方程和系统的分析构成了一个具有挑战性的新研究途径。在这样的微积分中，最小曲面是零平均曲率的超曲面，并且基本感兴趣的问题是著名的伯恩斯坦猜想的亚黎曼版本。后者与其经典祖先表现出明显的差异，并且存在许多与高斯图的奇点相关的新几何现象，这些现象通常发生在生成亚黎曼结构的子丛成为切空间的一部分的那些点上。超曲面。鉴于经典伯恩斯坦问题在上世纪数学发展中的重要作用，可以预见，次黎曼极小曲面理论及相应的伯恩斯坦问题将迎来广阔的发展。 PI 还建议寻找海森堡型群和 2 型西格尔域的 Folland-Stein 嵌入中的最小化器，从而计算最佳常数。该程序有助于解决更高余维 CR 流形的紧凑 CR Yamabe 问题。关于 CR Yamabe 问题，PI 建议研究由 Schoen 和 Yau 提出的相对论正质量定理的 CR 版本。预计前面提到的最小曲面理论将发挥重要作用。亚黎曼几何中的另一个新兴理论是蒙日-安培型方程，由于其与质量传递问题的紧密联系，它在几何学以及变分学中占据着中心地位。 PI 提议研究与 Alexandrov、Bakelman 和 Pucci 的几何极大值原理的亚黎曼版本相关的新估计。在联合工作中，他最近获得了适当类别的“凸”函数的结果，并且受到 N.Krylov 方法的启发，为涉及对称水平 Hessian 矩阵以及一些适当的交换子的函数建立了单调性类型结果。这个建议是卡诺群之间拟正则映射研究中出现的非线性方程的最优正则性。这是目前一个基本的悬而未决的问题，如果没有它的解决方案，就不可能取得实质性进展。在这方面，PI 还计划分析基本解和格林函数的唯一性这一棘手问题，并研究其水平集的几何性质。亚椭圆方程及其相关热流的边值问题（狄利克雷、诺伊曼）的研究、自由边界问题的研究以及几何和数学物理中出现的某些偏微分方程解的全局性质分析。偏微分方程及其形成的系统是描述大多数自然现象的基本规律。对物理世界的理解还需要掌握后者各种形式的基本几何结构。目前的建议属于研究的主流，它位于偏微分方程和系统理论（主要是非线性类型）及其与新兴几何类型（称为亚黎曼几何）的联系的交汇处。这两种理论在过去十年中都引起了人们的极大兴趣，并且继续吸引着国内外各数学家的兴趣。该提议还涉及数学物理和几何中的问题，其中对称性起着重要作用。对称性在自然界中无处不在，一个显着的例子是万有引力和静电引力的基本定律。研究自然现象产生对称性的条件对于实际后果及其对加深我们的知识的影响都很重要。