Sunghan Kim: Constraint maps and free boundaries

Abstract:

Constraint maps are critical points to the Dirichlet energy subject to an image constraint. Namely, the constraint restricts these maps to lie outside a given (smooth, open-connected) set in the target space, which can then be thought of as an obstacle. The target space is multi-dimensional, making this problem a natural extension of the obstacle problem into the vectorial setting. The presence of the obstacle naturally induces free boundary, which is the interface between the set where the image of the maps lies away from the obstacle and the set where their image lies on the obstacle. Yet, constraint maps also develop singularities, such as discontinuous points and branch points which have been the main subjects in the theory of harmonic maps and minimal surfaces, respectively. In this talk, I will discuss the recent breakthroughs on the tantalizing interplay between the free boundary and the mapping singularities for energy-minimizing constraint maps, based on a series of my collaboration with Alessio Figalli, André Guerra (ETH), and Henrik Shahgholian (KTH).