Existence, regularity and uniqueness results of geometric variational problems

几何变分问题的存在性、规律性和唯一性结果

• 批准号: 339133928
• 负责人: Professor Dr. Jonas Hirsch
• 金额: --
• 依托单位: Abteilung Analysis
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/339133928?language=en
• 关键词: Existence; regularity; uniqueness; results; geometric

Regularity and existence question arise naturally in the filed of geometric variational problems. This projects addresses some of them. Although regularity question are local in nature, some of them have global effects. We are particular interested in these once. Plateau’s problem of finding a “minimal surface” with a given boundary, had been very inspiring for mathematics. It lead to a variety of beautiful approaches. We will consider two of them, integer rectifiable area minimising currents and rectifiable area minimising currents mod(p). The latter one are rectifiable currents with multiplicity taking values in the integers mod(p). They are of interest since they allow for certain types of singularities. For instance we want to address the optimal boundary regularity for two dimensional area minimising currents, which has an immediate effect on the topology of a minimiser. Hence to give an answer is a major open problem in the field. And we want to investigate the local structure of the singular set of area minimising currents mod(p).At first we want to restrict ourselves to p odd and codimension one. An answer would give new insights into to structure of currents mod(p). It would hopefully revitalise the field.Polyconvex integrands play an important role in the calculus of variation. They arise naturally in mathematical models in elasticity. We want to investigate the discrepancy between a local regularity result and the existence of very wild solutions. Since on the one hand there is a local regularity result for minimisers on the other hand there are high oscillatory solutions obtained by convex integration. Any better understanding is of great interest.The Willmore energy is a well known geometric surface energy with applications in applied sciences. Beside others we are want to show in a general existence result for un-oriented minimisers.

规律性和存在性问题自然出现在几何变分问题中，尽管规律性问题本质上是局部性的，但其中一些问题具有全局影响。给定边界的“最小表面”对数学非常有启发，我们将考虑其中的两种方法，即整数可整流面积最小化电流和可整流面积最小化电流 mod(p)。后一种是具有多重性的可整流电流，其值取整数 mod(p)。它们很有趣，因为它们允许某些类型的奇点，例如，我们想要解决二维区域最小化电流的最佳边界规律性。对最小化器的拓扑有直接影响，因此给出答案是该领域的一个主要开放问题，我们想要研究奇异面积最小化电流 mod(p) 的局部结构。将我们限制在 p 奇数和余维 1 上。答案将为电流 mod(p) 的结构提供新的见解。多凸被积函数在变分计算中发挥着重要作用，它们在数学中自然出现。我们想要研究局部正则性结果与非常野生解的存在之间的差异，因为一方面存在极小化的局部正则性结果，另一方面存在高值。通过凸积分获得的振荡解是非常有意义的。威尔莫尔能量是一种众所周知的几何表面能量，在应用科学中有着广泛的应用。