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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.

当研究某一类几何对象时，特别注意与它们自然相关的量的极小或极大值通常是有帮助的。例如，在所有具有固定边界的曲面中，面积最小的曲面具有特殊的兴趣。对于这个项目，我们考虑了不同的最小化原则。它植根于流形理论，流形是曲面的高维等价。所讨论的几何对象是一个流形到另一个流形的映射。被最小化的量可以被认为是在这样的映射下曲率行为的量度。这种极小化问题的解称为双调和映射，它们产生了非线性偏微分方程组。变分法是数学分析中的一种理论，它为研究极小化问题和相应的微分方程提供了工具和方法。然而，双调和映射的问题并不完全符合这一理论的通常框架。这个项目的主要目标是使变分的方法与手头的问题的结构相一致。为此，几何对象的空间将不得不适当地扩展。该问题的偏微分方程组必须在扩展的空间上进行研究，所使用的解析方法必须与基础几何相结合，双调和映射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，另一些来自物理或其他领域的数学模型。对于双调和映射所得到的结果可能在其他理论中得到应用，反之亦然。因此，研究不会局限于双调和映射，尽管双调和映射是双调和映射的核心。目前对这类问题的变分方面了解不多。对这些问题的成功研究可能会为研究双调和地图和相关理论的几何学家或数学物理学家提供强大的变分工具。此外，它还将为变分和偏微分方程组理论增加新的方法。