The Variational Approach to Biharmonic Maps

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最小值或最大值通常是有帮助的。例如，在所有具有固定边界的曲面中，最小化面积的曲面是特别感兴趣的，对于这个项目，我们考虑不同的最小化原理。它植根于流形理论，流形理论是曲面的高维等价物。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度量。这种极小化问题的解称为双调和映射，它们产生非线性偏微分方程。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。这个项目的主要目的是调和变分法的方法与手头的问题的结构。为此，几何对象的空间将不得不适当地扩展。问题的偏微分方程必须在扩展空间上研究，为此目的所用的分析方法必须与基础几何相结合。双调和映射问题只是具有类似结构的几个问题之一。它们中的一些是从几何学中推导出来的，另一些是从物理学或其他领域的数学模型中推导出来的。双调和映射的结果很可能在其他理论中找到应用，反之亦然。因此，研究将不限于双调和映射，虽然他们是在其中心。目前还不知道多少变分方面的问题，这类。对这些问题的成功研究将使变分法成为研究双调和映射及其相关理论的几何学家或数学物理学家的有力工具。此外，它将增加新的方法，变分法和偏微分方程的理论。

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.