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中的自然问题。这个项目解决了其中的一些问题。虽然正则性问题本质上是局部的，但其中一些问题具有全局影响。我们曾经对这些特别感兴趣。高原的问题，找到一个“最小的表面”与给定的边界，一直非常鼓舞人心的数学。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，其多重性取整数mod（p）中的值。它们是有趣的，因为它们允许某些类型的奇点。例如，我们想解决二维面积最小化电流的最佳边界正则性，这对极小化器的拓扑结构有直接影响。因此，给出答案是该领域的一个重大开放问题。我们想研究奇异面积最小化流集mod（p）的局部结构，首先我们想把自己限制在p奇数和余维1。一个答案将给新的见解，以结构的电流模（p）。多凸被积函数在变分学中起着重要的作用。它们在弹性力学的数学模型中自然出现。我们要调查的差异之间的局部正则性结果和存在非常野生的解决方案。由于一方面有一个局部正则性的结果，另一方面，有高振荡的解决方案获得凸积分。任何更好的理解都是非常有趣的。Willmore能量是一种众所周知的几何表面能，在应用科学中有应用。除此之外，我们还想证明非定向极小元的一般存在性结果。

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.