Analysis and PDE Seminar: Fine properties of Dirichlet energy minimizing multi-valued functions

Seminar | November 27 | 4:10-5 p.m. | 740 Evans Hall

 Brian Krummel, UC Berkeley

 Department of Mathematics

I will discuss the fine structure of the branch set of multivalued Dirichlet energy minimizing functions as developed by Almgren. It is well-known that the dimension of the interior singular set of a Dirichlet energy minimizing function on an $n$-dimensional domain is at most $n-2$. We show that the singular set is countably $(n-2)$-rectifiable and also prove the uniqueness of homogeneous tangent functions at almost every singular point. Our approach involves adapting a “blow up” method due to Leon Simon, which was originally applied to multiplicity one classes of minimal submanifolds. We apply Simon’s method in the higher multiplicity setting of multivalued energy minimizers using techniques from prior work of Neshan Wickramasekera together with new estimates. This is joint work with Neshan Wickramasekera.