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.

主要研究者：Nicola Garofalo，Purdue UniversityDMS-0300477摘要：在过去的世纪中，分析和几何的发展受到了解决各种基本问题的愿望的极大影响，这些问题涉及到一些特殊的偏微分方程，大多是非线性类型的。虽然大多数这些问题现在已经解决了在经典的欧几里得或黎曼设置，他们的次黎曼对应目前形成一个机构的基本开放的问题。这项建议的一个更广泛的目标是研究其中的一些问题。这个PI关注的是发展一个新的理论的极小曲面，或更一般的曲面有界平均曲率，在次黎曼空间，研究其正则性和分类的等周集在一些模型空间的对称性。他提出了一种基于水平高斯映射思想的超曲面微积分，并由此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条具有挑战性的新的研究途径。在这样的微积分中，极小曲面是平均曲率为零的超曲面，一个基本的问题是著名的伯恩斯坦猜想的次黎曼版本。后者显示了显着的差异，其经典的祖先，有一个主机的新的几何现象与奇异的高斯映射一般发生在这些点的子丛，产生的子黎曼结构成为切空间的一部分，以超曲面。鉴于经典的伯恩斯坦问题在上个世纪数学发展中所起的作用，可以预见，亚黎曼极小曲面理论及相应的伯恩斯坦问题将有广阔的发展前景。PI还提出在海森堡型群和2型Siegel域的Folland-Stein嵌入中找到最小值，从而计算最佳常数。该程序是解决高余维CR流形的紧CR Yamabe问题的工具。关于CR Yamabe问题，PI建议研究Schoen和Yau的相对论正质量定理的CR版本。可以预料，前面提到的极小曲面理论将发挥重要作用。在次黎曼几何中另一个新兴的理论是蒙格-萨菲尔类型的方程，它在几何学和变分法中占据中心地位，因为它们与质量输运问题紧密相连。PI建议研究一个新的估计与亚黎曼版本的几何最大值原理的亚历山德罗夫，巴克尔曼，和普奇。在联合工作中，他最近获得的结果适当的类“凸”的功能，并受到N.克雷洛夫的方法，建立单调型结果的功能涉及对称化水平海森沿着与一些适当的administrators。另一个问题包括在这个建议是最佳的非线性方程的正则性所产生的研究拟正则映射之间的卡诺集团。这是目前一个基本的悬而未决的问题，没有它的解决方案，这将是不可能取得实质性进展的非线性潜力理论的次黎曼空间。在这方面，PI还计划分析基本解和绿色函数的唯一性的微妙问题，并研究其水平集的几何性质。其他方向的调查是分析边界值问题（狄利克雷，诺依曼）的次椭圆方程及其相关的热流，研究自由边界问题，并分析全球性质的解决方案，以一些偏微分方程的出现在几何和数学物理。偏微分方程和由后者构成的系统是描述大多数自然现象的基本定律。理解物理世界还需要掌握物理世界各种形式的基本几何结构。目前的建议属于主流的研究，它坐落在汇合的理论偏微分方程和系统，主要是非线性类型，以及它们的连接与新兴类型的几何，称为亚黎曼几何。这两个理论都见证了爆炸的兴趣在过去的十年中，他们继续吸引各种学校的数学家在全国和国外的兴趣。这个建议也涉及数学物理和几何问题，其中对称性起着重要作用。对称性在自然界中无处不在，一个显著的例子是引力和静电引力的基本定律。研究一种自然现象发展对称性的条件，对于实际结果和它对我们知识的深化的意义都是重要的。