Sunghan Kim: On vectorial obstacle problems

Abstract:

In this talk, I will discuss some new observations on vectorial obstacle problems. The problem itself has a long history, which began around late 70s and continued through 80s. However, the problem was studied from the standpoint of the theory of harmonic maps, and the main focus was laid upon the partial regularity of solutions, rather than their behavior near the free boundaries. About three decades later, the extension of free boundary problems into vectorial settings has regained its attention from the free boundary community. Very recently, we observed some interesting issues regarding the behavior of the vectorial solutions. More specifically, when we decompose the solutions into tangential and normal components, the optimal regularity of the tangential component has been untouched, and turned out to be highly tantalizing. The study of the optimal regularity for the tangential component seems to open up some new type of PDE in connection with the free boundary problems, which is interesting of its own. In this talk, I will present some partial result obtained in this direction. This talk will be based on a recent collaboration with A. Figalli and H. Shahgholian.