Some nonlinear problems in analysis and geometry

摘要：解析学和几何学在过去一个世纪的发展，很大程度上受到了解决涉及一些特殊偏微分方程的各种基本问题的愿望的影响，这些问题大多是非线性的。虽然这些问题中的大多数现在已经在经典的欧几里得或黎曼环境中得到解决，但它们的亚黎曼环境目前形成了一系列基本的开放性问题。本提案的一个更广泛的目标是研究其中的一些。这个PI是关于在亚黎曼空间中发展最小曲面的新理论，或者更一般的有界平均曲率曲面，研究它们的正则性，并对一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映射思想的超曲面微积分，并提出了平均曲率的新概念，由此产生的非线性方程和系统的分析构成了一个具有挑战性的研究新途径。在这样的微积分中，最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亚黎曼版本。后者显示出与其经典祖先的显著差异，并且有许多新的几何现象与高斯映射的奇点有关，这些奇点通常发生在产生亚黎曼结构的子束成为超曲面切线空间的一部分的点上。鉴于经典Bernstein问题在上个世纪数学发展中的作用，可以预见，亚黎曼极小曲面理论和相应的Bernstein问题将会焕发出广阔的发展前景。PI还提出了在Heisenberg型群和类型2的Siegel域群的Folland-Stein嵌入中寻找最小值，从而计算出最佳常数。该程序对解决高余维CR流形的紧化CR Yamabe问题具有重要意义。关于CR - Yamabe问题，PI提议从Schoen和Yau的相对论中研究正质量定理的CR版本。预计前面提到的最小曲面理论将发挥重要作用。亚黎曼几何中另一个新兴的理论是蒙日-安普雷型方程，它在几何和变分学中占据中心地位，因为它与质量输运问题有着密切的联系。PI建议研究与Alexandrov， Bakelman和Pucci的几何极大原理的亚黎曼版本相关的新估计。在最近的联合工作中，他得到了适当的“凸”函数类的结果，并受到N.Krylov方法的启发，建立了涉及对称水平Hessian和一些适当的换向子的泛函的单调型结果。本文提出的另一个问题是在研究卡诺群之间的拟正则映射时产生的非线性方程的最优正则性问题。这是目前一个基本的开放性问题，没有它的解决方案，就不可能在亚黎曼空间的非线性势理论中取得实质性的进展。在这方面，PI还计划分析基本解和格林函数的唯一性这一微妙问题，并研究它们的水平集的几何性质。其他研究方向是分析亚椭圆方程的边值问题（Dirichlet, Neumann）及其相关的热流，研究自由边界问题，以及分析几何和数学物理中出现的一些偏微分方程解的整体性质。偏微分方程和由后者形成的系统是描述大多数自然现象的基本定律。对物理世界的理解还需要掌握后者的各种形式的潜在几何结构。目前的建议属于研究的主流，它位于偏微分方程和系统的理论，主要是非线性类型，以及它们与一种新兴的几何类型，称为亚黎曼几何的联系的汇合。在过去的十年里，这两种理论都引起了人们的兴趣，并继续吸引着国内外各种数学流派的兴趣。这一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。对称性在自然界无处不在，一个显著的例子就是万有引力和静电吸引的基本定律。研究自然现象形成对称性的条件，对于实际结果和对我们知识的进一步发展都是重要的。

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.