The Variational Approach to Biharmonic Maps

双调和映射的变分方法

基本信息

批准号：
EP/F048769/1
负责人：
Roger Moser
金额：
$ 27.52万
依托单位：
University of Bath
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FF048769%2F1
关键词：
Variational Approach Biharmonic Maps

项目摘要

When studying a certain class of geometric objects, it is often helpful to give special attention to the minimizers or maximizers of the quantities naturally associated to them. For instance, among all surfaces with fixed boundary, the ones that minimize the area are of special interest.For this project, we consider a different minimization principle. It is rooted in the theory of manifolds, which is the higher-dimensional equivalent of surfaces. The geometrical objects in question are mappings of one manifold onto another. The quantity that is to by minimized, can be thought of as a measure for the behaviour of the curvature under such a mapping. The solutions of this minimization problem are called biharmonic maps, and they give rise to nonlinear partial differential equations.The calculus of variations is a theory from mathematical analysis which provides tools and methods to study minimization problems and the corresponding differential equations. The problem of biharmonic maps, however, does not quite fit into the usual framework of this theory. The main object of this project is to reconcile the methods of the calculus of variations with the structure of the problem at hand.To this end, the space of geometrical objects will have to be extended appropriately. The partial differential equations of the problem have to be studied on the extended space, and the analytic methods used for this purpose have to be combined with the underlying geometry.The problem of biharmonic maps is only one of several problems with similar structures. Some of them are derived from geometry, others from mathematical models in physics or other fields. Results obtained for biharmonic maps are likely to find applications in one of the other theories, and vice versa. Therefore the research will not be limited to biharmonic maps, although they are at its centre.Not much is currently known about variational aspects of problems of this type. A successful study of these questions could make the powerful tools of the calculus of variations available to geometers or mathematical physicists studying biharmonic maps and related theories. In addition, it will add new methods to the theories of the calculus of variations and partial differential equations.

在研究某种类别的几何对象时，特别注意与它们自然相关的数量的最小化器或最大化器通常会有所帮助。例如，在所有具有固定边界的表面中，最小化该区域的边界是特别感兴趣的。对于此项目，我们考虑了不同的最小化原则。它植根于流形理论，这是表面的较高维度。所讨论的几何对象是将一个歧管映射到另一个歧管上。可以将最小化的数量视为在这种映射下曲率行为的量度。这个最小化问题的解决方案称为Biharmonic图，它们引起了非线性偏微分方程。变异的计算是一种来自数学分析的理论，它提供了研究最小化问题和相应微分方程的工具和方法。但是，Biharmonic地图的问题并不完全适合该理论的通常框架。该项目的主要对象是将变化的计算方法与手头问题的结构进行调解。为此，必须适当地扩展几何对象的空间。问题的部分微分方程必须在扩展空间上进行研究，并且用于此目的的分析方法必须与基础几何形状结合使用。BiharmonicMaps的问题只是具有相似结构的几个问题之一。其中一些来自几何形状，另一些源自物理或其他领域的数学模型。 Biharmonic图获得的结果可能会在其他理论之一中找到应用，反之亦然。因此，该研究不仅限于Biharmonic地图，尽管它们处于中央。对这些问题的成功研究可能会使可用的变体计算的强大工具可用于几何或数学物理学家研究Biharmonic地图和相关理论。另外，它将为变异和部分微分方程的计算理论添加新方法。