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

抽象卡波尼亚 这个项目的主题是研究内部和边界 正则性极小的泛函，涉及"水平"梯度的功能定义在卡诺集团或在一个子黎曼流形。 这样的泛函在几何和偏微分方程中有着特殊的意义。 微分方程，包括能量泛函和面积泛函。在正则性理论的应用中， 一个是次拉普拉斯算子的狄利克雷问题，正则性和刚性 卡诺群与几何之间拟共形映射的性质 最小的表面。作为主题的一种变化，这个项目也 包括研究适当定义的能量映射的极小化 亚黎曼目标 变分问题极小解的正则性研究 几个世纪以来一直是数学中的主要问题之一。 它与自然现象的研究有着密切的联系，因为在自然界中， 能量最小化配置最有可能出现。 如果一个极小化子是非常正则的，那么研究它的结构就很容易。 从数学形式上来说，这意味着物理系统对应于一个非常规则的极小化 可以非常准确地预测。这种观测是从气象预报到航天飞机发射计划（取决于具体功能）等广泛应用的基础。 最小化研究）。 该项目中涉及的变分问题与空间的特殊结构有关，这种空间结构在许多物理系统的研究中自然产生，其中包括晶体系统和晶体材料。这种特殊结构的一个主要特点是， 空间中的方向是相等的。沿着一些 而不是沿着沿着移动。与此相关的数学问题 功能是非常困难和有趣的。

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.