Non-linear equations in analysis and geometry

分析和几何中的非线性方程

• 批准号: 0070492
• 负责人: Nicola Garofalo
• 金额: $ 17.7万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070492&HistoricalAwards=false
• 关键词: Non; linear; equations; analysis; geometry

This proposal is concerned with two main projects. The former focuses onvarious questions in sub-Riemannian geometry and in the closely connectedanalysis of sub-elliptic pde's and systems. The PI proposes to investigatethe classification of non-negative entire solutions to non-linearequations in groups of Heisenberg type, and compute the best constants inthe Folland-Stein Sobolev embedding. This program is instrumental to apossible attack of the compact CR Yamabe problem in the open case of CRmanifolds of co-dimension higher than one. The geometric case of suchembedding will also be investigated along with the relative isoperimetricinequalities. The PI also proposes to study the regularity of minimalsurfaces, the question of traces on lower dimensinal sub-manifolds offunctions having integrable horizontal derivatives. The basic boundaryvalue problems, such as the Dirichlet and the Neumann problem will also beinvestigated, and a theory of variational inequalities and regularity of"free boundaries" will be developed. The second project is concerned withvarious problems in which symmetry plays an important role. One of them isconcerned with the determination of the extremal functions in theTomas-Stein restriction theorem for the Fourier transform. Other problemsare connected with symmetry in the exterior obstacle problem, a conjectureof De Giorgi connected to minimal surfaces, and symmetry in the evolutionof surfaces driven by mean curvature.Partial differential equations and systems formed by the latter are thebasic laws which describe most natural phenomena. An understanding of thephysical world also requires grasping the underlying geometric structureof the latter in its various forms. The present proposal belongs to thatmainstream of research which sits at the confluence of the theory ofpartial differential equations and systems, both linear and non-linear,and their connections with an emerging type of geometry, calledsub-Riemannian geometry. Both theories have witnessed an explosion ofinterest in the last decade and they continue to attract the interest ofvarious schools of mathematicians both nationwide and abroad. Another mainpart of this proposal is devoted to the study of physical and mathematicalproblems in which symmetry plays an important role. Symmetry is presenteverywhere in nature, a remarkable instance being the fundamental lawsof gravitation and electrostatic attraction. The study of conditions underwhich a given natural phenomenon develops symmetries is both important forits practical consequences (the presence of symmetriesdrastically reduces the human effort) and for its implicationsin the furthering of our knowledge.

该提案涉及两个主要项目。前者侧重于次黎曼几何中的各种问题以及次椭圆偏微分方程和系统的密切相关分析。 PI 提议研究海森堡型群中非线性方程组非负整解的分类，并计算 Folland-Stein Sobolev 嵌入中的最佳常数。该程序有助于在余维大于 1 的 CR 流形的开放情况下解决紧凑 CR Yamabe 问题。还将研究这种嵌入的几何情况以及相对等周不等式。 PI还建议研究最小曲面的正则性，即具有可积水平导数的函数的低维子流形上的迹问题。还将研究基本边值问题，例如狄利克雷和诺伊曼问题，并发展变分不等式和“自由边界”正则性理论。第二个项目涉及对称性起着重要作用的各种问题。其中之一涉及傅立叶变换托马斯-斯坦限制定理中极值函数的确定。其他问题与外部障碍问题中的对称性、与最小曲面相关的德乔治猜想以及由平均曲率驱动的曲面演化中的对称性有关。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各种形式的基本几何结构。本提案属于主流研究，该研究融合了线性和非线性偏微分方程和系统理论及其与新兴几何类型（称为亚黎曼几何）的联系。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流派数学家的兴趣。该提案的另一个主要部分致力于研究对称性起着重要作用的物理和数学问题。对称性在自然界中随处可见，一个显着的例子是万有引力和静电引力的基本定律。对给定自然现象产生对称性的条件的研究不仅对于其实际后果（对称性的存在大大减少了人类的努力）而且对于进一步加深我们的知识的影响都很重要。