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.

规律性和存在的问题自然而然地提出了几何变异问题。该项目解决了其中的一些。尽管规律性问题本质上是本地性的，但其中一些具有全球影响。我们曾经对这些特别感兴趣。高原找到具有给定边界的“最小表面”的问题，对数学非常鼓舞人心。这导致了各种美丽的方法。我们将考虑其中的两个，整数可纠正的区域最小化电流和最小化电流mod（P）的区域。后一个是可整流的电流，在整数mod（p）中采用多重性。它们很感兴趣，因为它们允许某些类型的奇点。例如，我们希望解决最小化电流的最佳边界规则性，这对最小值的拓扑产生了直接影响。因此，给出答案是该领域的主要开放问题。我们想研究最小化电流mod（p）的奇异区域集合的局部结构。首先，我们希望将自己限制在p奇数和编成imimension上。答案将使对电流模型结构（P）的结构提供新的见解。有望振兴该领域。PolyConvexIntegrateds在变化的计算中起着重要作用。它们自然是弹性中的数学模型。我们想研究当地规律性结果与存在非常野生解决方案之间的差异。由于一方面，最小化剂的局部规律性结果是通过凸积分获得的高振荡溶液。任何更好的理解都引起了人们的极大兴趣。WillmoreEnergy是众所周知的几何表面能量，并在应用科学中应用。除其他人以外，我们还想在不面向的最小化器的一般存在结果中展示。