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.

这项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组和系统的紧密联系的分析。PI建议在Heisenberg型群中研究非线性方程的非负整体解的分类，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型流形的开放情形下对紧致CR Yamabe问题进行可能的攻击。我们还将研究这种夹层的几何情况以及相对等周不等式。PI还建议研究极小曲面的正则性，即具有可积水平导数的函数的低维子流形上的迹问题。基本边值问题，如Dirichlet问题和Neumann问题，也将被研究，并将发展一个变分不等式和“自由边界”的正则性理论。第二个项目涉及各种问题，其中对称性起着重要作用。其中之一是关于傅里叶变换的Tomas-Stein限制定理中极值函数的确定。其他问题还涉及外部障碍问题中的对称性、极小曲面上的De Giorgi猜想以及由平均曲率驱动的曲面演化中的对称性。由后者形成的偏微分方程组和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基本几何结构。目前的建议属于研究偏微分方程组和系统理论的主流，包括线性和非线性的，以及它们与一种被称为次黎曼几何的新兴几何类型的联系。在过去的十年里，这两个理论都见证了人们的兴趣的爆炸性增长，它们继续吸引着国内外不同学派的数学家的兴趣。这项建议的另一个主要部分致力于研究对称性在其中起重要作用的物理和数学问题。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吸引的基本定律。研究一种给定的自然现象产生对称性的条件，既是因为它的实际结果(对称性的存在极大地减少了人类的努力)，也是因为它涉及到我们知识的进一步发展。