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

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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.