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

9706892 Garofalo 本计划涉及三个方面：1）Carnot-Caratheodory（CC）空间中非线性方程组和方程组的最优正则性及其在几何、等周不等式和Sobolev不等式、（CC）空间中极小曲面及其正则性、幂零李群中的Dirichlet问题、调和测度与周长的关系、CR流形间调和映射等方面的应用。2)超定边值问题的对称性，包括内部和外部，在欧几里得几何和CR几何中，在等周和Sobolev不等式中的极小化，在Heisenberg群中，E.德·乔治和它的抛物线对应物。3)次椭圆算子的唯一延拓，散射的应用，海森堡群上的反问题，非线性p-Laplacian型方程的唯一延拓。 拟议中的研究位于数学中两个主要领域的交汇处，即偏微分方程和几何。这两个领域都在观察和描述自然现象中出现的问题中找到了它们的起源和动机，在每一个尺度上。拟议研究的一个主要重点是，例如，物体形状的重建知道一些可以在物体周围的表面上测量的量。这样一个问题有很大的相关性，在应用科学，特别是在联邦的战略利益，从计算机断层扫描，生物技术，控制核反应堆的核心等领域的重要性，拟议的研究也将有助于通过年轻的研究人员（博士生）的参与人力资源的发展。首席研究员还在写一本书，重点介绍过去几年与几名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.