Sunghan Kim: Constraint map heat flows

This talk will be concerned with constraint map heat flows (CMHFs), which can be viewed as a natural extension of the parabolic obstacle problem to the vectorial setting. In a nutshell, this is the negative gradient flow within the class of maps whose images are constrained to lie outside a prescribed set, which acts as an obstacle. CMHFs were introduced in the late eighties with a global existence result parallel to that for harmonic map heat flows. However, very little has been known about their behavior near the free boundary, which is defined as the topological boundary of the non-coincidence set. In this talk, in addition to discussing the regularity of the free boundary, we will present an example of a heat flow that blows up in finite time at a free boundary point. This provides a striking contrast with the positive regularity results for energy-minimizing constraint maps.