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解的特定映射通常是有帮助的。这是调和映射作为狄利克雷泛函的临界点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，但是这个问题的几个变体仍然会引起一些基本性质的开放性问题。这些包括带有额外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些函数的正则性以及在某些边界条件或其他边条件下的存在性。对于正则性问题，以前类似问题的工作使用了单调性公式，需要将其推广到所研究的问题中。下一步很可能是基于Riviere的想法，他在合适的规范变换的帮助下重写了类似的方程，这样它们就可以在某些函数空间中进行估计。对于本项目研究的问题，该方法也需要一些扩展。存在性问题对于双谐波/多谐波地图来说尤其有趣。潜在的困难是由于可能的能量集中，能量泛函缺乏矫顽力。进展将需要在这里详细分析这种集中。

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.