Sunghan Kim: Constraint maps with free boundaries

Constraint maps are the maps that minimize the Dirichlet energy under some constraints on their image. The study of constraint maps dates back to the 1970s. As harmonic maps, these maps can be discontinuous, and for this reason, the early development for the constraint maps was focused on the partial regularity. In contrast (to harmonic maps), constraint maps give rise to free boundaries, which are of obstacle-type. Despite its long history, their free boundaries were studied only at a basic level. However, recent breakthroughs in the (scalar) obstacle problems seem to open up many new and interesting issues regarding the fine properties of these maps. In this talk, I will briefly discuss the history of the constraint maps, the connection to the (scalar) obstacle problems, and present a very recent result on some fine properties of these maps. The talk will be based on a recent joint work with Alessio Figalli (ETH) and Henrik Shahgholian (KTH).