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

这个项目的主要主题是研究涉及定义在卡诺群或次黎曼流形中的函数的“水平”梯度的泛函的极小元的内部和边界正则性。这种泛函对几何方程和偏微分方程都有特别的兴趣，包括能量泛函和面积泛函。在正则性理论的应用中，有次拉普拉斯的Dirichlet问题，卡诺群与极小曲面几何之间拟共形映射的正则性和刚性性质。作为主要主题的变体，本项目还包括对具有次黎曼目标的地图的适当定义的能量最小化的研究。几个世纪以来，对变分问题极小化正则性的研究一直是数学中的主要问题之一。它与自然现象的研究密切相关，因为在自然界中，最有可能出现的是能量最小化构型。如果极小元是非常正则的，那么很容易研究它的结构。从数学形式翻译出来，这意味着可以非常准确地预测与非常规则的极小值相对应的物理系统。从气象预报到航天飞机发射的规划(取决于所研究的具体功能和最小化)，这种观测是广泛应用的基础。这个项目中涉及的变分问题与空间的一种特殊结构有关，这种结构是在研究许多物理系统时自然产生的，其中包括热力学系统和晶体材料。这种特殊结构的一个主要特点是，并不是空间中的所有方向都是等价的。沿着一些方向旅行的成本比沿着其他方向移动的成本更高。与此功能相关的数学问题既难又有趣。

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.