Non-linear equations in analysis and geometry

这项建议涉及两个主要项目。前者着重于亚黎曼几何中的各种问题以及与之密切相关的亚椭圆偏微分方程和系统的分析。PI提出研究Heisenberg型群中非线性方程的非负整解的分类，并计算Folland-Stein Sobolev嵌入中的最佳常数。这个程序是有助于apossible攻击的紧CR Yamabe问题在开放的情况下，CR流形的余维大于1。这样的矩阵的几何情形也将沿着与相对的等周不等式一起被研究。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的问题。基本的边值问题，如Dirichlet和Neumann问题也将被研究，变分不等式理论和“自由边界”的正则性将被发展。第二个项目是关于对称性起重要作用的各种问题。其中之一是关于Fourier变换的Tomas-Stein限制定理中极值函数的确定。其他问题与对称性有关的有外障碍问题、与极小曲面相连的De Giorgi定理以及由平均曲率驱动的曲面演化中的对称性，后者所形成的偏微分方程和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，它坐落在汇合理论的偏微分方程和系统，线性和非线性，以及它们的连接与新兴类型的几何，所谓的亚黎曼几何。这两个理论在过去的十年里都引起了人们极大的兴趣，并继续吸引着国内外各个数学流派的兴趣。这个建议的另一个主要部分是致力于研究物理和物理问题，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。研究一个给定的自然现象发展对称性的条件，对于它的实际结果（对称性的存在大大减少了人类的努力）和它在促进我们知识方面的意义都是重要的。

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.