Variational problems for maps between manifolds

流形之间映射的变分问题

• 批准号: 2748267
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2748267
• 关键词: Variational; problems; maps; between; manifolds

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.

为了理解流形的几何形状以及它们之间的映射，研究作为某些变分问题的解而出现的特定映射通常是有帮助的。这就是调和映射（狄利克雷泛函的关键点）得到广泛研究的原因之一。这个具体问题现在已经很好地理解了，但是这个问题的几个变体仍然会引起具有相当基本性质的开放问题。其中包括带有额外项的调和映射方程或双调和/多调和映射。该项目的目的是研究这些规律性以及在某些边界条件或其他边条件下的存在性。对于规律性问题，以前对类似问题的工作使用了单调性公式，需要将其扩展到所研究的问题。下一步很可能是基于里维埃的想法，他在适当的规范变换的帮助下重写了类似的方程，以便它们允许在某些函数空间中进行估计。对于本项目中研究的问题，该方法也需要一些扩展。对于双调和/多调和映射来说，存在性问题尤其有趣。潜在的困难是由于可能的能量集中而导致能量泛函缺乏矫顽力。进展需要在这里对此集中进行详细分析。