Existence, regularity and uniqueness results of geometric variational problems

• Applicant: Professor Dr. Jonas Hirsch
• Subject Area: Mathematics
• Term: from 2017 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 339133928

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity