Boundary and Interior Regularity for Minimizers of the p-Energy and Related Functionals in Carnot-Caratheodory Spaces

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Abstract Capogna The main theme of this project is the study of the interior and boundary regularity of minimizers for functionals that involve the "horizontal" gradient of functions defined in a Carnot group or in a sub-Riemannian manifold. Such functionals have a special interest both in geometry and partial differential equations, and include the energy functional and the area functional. Among the applications of the regularity theory one has the Dirichlet problem for subLaplacians, regularity and rigidity properties for quasiconformal maps between Carnot groups and the geometry of minimal surfaces. As a variation on the main theme, this projects also includes the study of minimizers for a suitably defined energy of maps with sub-Riemannian target. The study of the regularity for minimizers of variational problems has been one of the main concerns in mathematics for centuries. It is deeply related to the study of natural phenomenon, since in nature the energy minimizing configurations are the most likely to appear. If a minimizer is very regular then it is easy to study its structure. Translated out of the mathematical formalism, this means that the physical system corresponding to a very regular minimizer can be predicted very accurately. This observation is at the basis of a wide range of applications, from meteorological forecasts to the planning of a space shuttle launch (depending on the specific functional and minimizer studied). The variational problems involved in this project are associated to a special structure of the space that arises naturally in the study of many physical systems, among them thermodynamical systems and crystalline materials. A main feature of this special structure is that not all directions in space are equivalent. It costs more to travel along some directions than to move along others. The mathematical problems related to this feature are very difficult and interesting.