Optimal Regularity for Nonlinear Pde's and Systems in Carnot-Caratheodory Spaces and Applications to Geometry, Symmetry for Pde's, Unique Continuation

本文主要研究了三个问题：1)Carnot-Caratheodory （CC）空间中非线性pde和系统的最优正则性及其在几何中的应用，等周不等式和Sobolev不等式，（CC）空间中的极小曲面及其正则性，幂零李群中的Dirichlet问题：调和测度与周长的联系，CR流形之间的调和映射。2)欧几里得几何和CR几何中内外超定边值问题的对称性，Heisenberg群中等周不等式和Sobolev不等式的最小化，E. De Giorgi及其抛物线对应物的一个猜想。3)次椭圆算子的唯一延拓，在散射中的应用，Heisenberg群上的逆问题，p- laplace型非线性方程的唯一延拓。拟议的研究位于数学中被称为偏微分方程和几何的两个主要兴趣领域的交汇处。这两个领域都在观察和描述各种尺度的自然现象时产生的问题中找到了它们的起源和动机。所提出的研究的一个主要焦点是，例如，在知道物体周围表面可以测量的一些量的情况下重建物体的形状。这样的问题在应用科学中具有很大的相关性，在涉及联邦战略利益的领域尤其重要，这些领域包括计算机断层扫描、生物技术、核反应堆堆芯的控制等。拟议的研究还将通过青年研究人员（博士生）的参与促进人力资源的发展。首席研究员也在写一本书，这本书将集中在过去几年里与几位博士生合作开发的项目的各个方面。

9706892 Garofalo The proposal is concerned with three projects: 1) Optimal regularity for nonlinear pde's and systems in Carnot-Caratheodory (CC) spaces and the applications of this theory to geometry, isoperimetric and Sobolev inequalities, minimal surfaces in (CC) spaces and their regularity, Dirichlet problem in nilpotent Lie groups: Connection between harmonic measure and perimeter, harmonic maps between CR manifolds. 2) Symmetry in overdetermined boundary value problems, both interior and exterior, in Euclidean and CR geometry, minimizers in the isoperimetric and Sobolev inequalities in the Heisenberg group, a conjecture of E. De Giorgi and its parabolic counterpart. 3) Unique continuation for sub-elliptic operators, applications to scattering, inverse problems on the Heisenberg group, unique continuation for nonlinear equations of p-Laplacian type. The proposed research sits at the confluence of two main areas of interest in mathematics known as partial differential equations and geometry. Both areas find their origin and motivation in problems arising in the observation and description of natural phenomena, at every scale. A main focus of the proposed research is, e.g., the reconstruction of the shape of a body knowing some quantities that can be measured on the surface that surrounds the body. Such a problem has a great relevance in the applied sciences and is especially important in areas of Federal strategic interest ranging from computerized tomography, to biotechnology, to control of the core of a nuclear reactor, etc. The proposed research will also contribute to the development of human resources through the involvement of young investigators (doctoral students). The principal investigator is also writing a book which will focus on those aspects of the program that has been developed over the past few years in collaboration with several doctoral students.