Variational problems for maps between manifolds

为了理解流形的几何和它们之间的映射，研究作为某些变分问题的解而产生的特定映射通常是有帮助的。这就是为什么作为Dirichlet泛函的临界点的调和映射被广泛研究的原因之一。这个具体的问题现在已经被很好地理解了，但这个问题的几个变种仍然引起了相当根本的开放问题。其中包括带有附加项的调和映射方程或双调和/多调和映射。本课题的目的是研究这类问题的正则性及其在一定边界条件或其他边值条件下的存在性，对于正则性问题，以前的工作是利用一个单调性公式，这个公式将需要推广到所研究的问题。下一步很可能是基于Riviere的想法，他借助适当的规范变换重写了类似的方程，以便它们允许在某些函数空间中进行估计。对于本项目中所研究的问题，该方法也需要一些扩展。存在问题首先对于双调和/多调和映射是有趣的。潜在的困难在于，由于可能的能量集中，能量泛函缺乏矫顽力。要取得进展，就需要在这里详细分析这种集中度。

In order to understand the geometry of manifolds and the maps between them, it is often helpful to study specific maps that arise as the solutions of certain variational problems. This is one reason why harmonic maps, which are the critical points of the Dirichlet functional, have been studied extensively. This specific problem is very well understood now, but there are several variants of the problem that still give rise to open problems of a quite fundamental nature. These include the harmonic map equation with an extra term or biharmonic/polyharmonic maps. The purpose of this project is to study the regularity of these and the existence under certain boundary conditions or other side conditions.For the regularity question, previous work for similar problems makes use of a monotonicity formula, which will need to be extended to the problems studied. The next step will most likely be based on ideas of Riviere, who rewrote similar equations with the help of suitable gauge transformations, so that they permit estimates in certain function spaces. For the problems studied in this project, this method will require some extensions, too.Existence questions are interesting above all for biharmonic/polyharmonic maps. The underlying difficulty is a lack of coercivity of the energy functional due to possible energy concentration. Progress will require a detailed analysis of this concentration here.