Existence, regularity and uniqueness results of geometric variational problems

正则性和存在性问题是几何变分问题的重要组成部分。这个项目解决了其中的一些问题。规律性问题虽然是局部性的，但有些问题具有全球性的影响。我们对这些特别感兴趣。Plateau的问题是寻找具有给定边界的“最小曲面”，这对数学来说是非常鼓舞人心的。它导致了各种各样的美丽的方法。我们将考虑其中的两个，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)。后者是可整流电流，其多重度取整数mod(p)的值。它们是有趣的，因为它们允许某些类型的奇点。例如，我们想要解决二维面积最小化电流的最佳边界规则，这对最小化器的拓扑结构有直接影响。因此，给出一个答案是该领域的一个主要开放性问题。我们想要研究面积极小电流模(p)奇异集的局部结构。一开始我们想限制在p奇和余维为1的情况下。一个答案将对电流的结构模型(p)提供新的见解。它有望使该领域重新焕发活力。聚凸积分在变分学中起着重要的作用。它们自然出现在弹性的数学模型中。我们要研究局部正则性结果与非常野解的存在性之间的差异。由于一方面存在极小值的局部正则性结果，另一方面存在由凸积分得到的高振荡解。任何更好的理解都是非常有趣的。威摩能是一种众所周知的几何表面能，在应用科学中有着广泛的应用。除此之外，我们还想展示非定向最小值的一般存在性结果。

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.