Sunghan Kim: Minimizing constraint maps

Abstract: This talk is concerned with constraint maps, where the theory of harmonic maps intersects with free boundary problems. The vectorial maps naturally develop discontinuous singularities for topological reasons, and they also give rise to free boundaries due to the constraints. The basic regularity results were established back in the 80s, but many important issues have been left uncovered to this day. Over the last couple of years, my collaborators and I revisited the problem, asking ourselves: “would the set of singularities meet the free boundaries?”, a central question yet highly tantalizing to answer. Very recently, we successfully tackled this issue for minimizing constraint maps. In this talk, I will provide some background for the mappings and go through our ideas in establishing the regularity issue near their free boundaries. My talk will be based on the joint works by Alessio Figalli (ETH), André Guerra (ETH) and Henrik Shahgholian (KTH).