Abstract: In this talk, we consider the maximizing problem associated with Sobolev embedding related to the space of bounded variation of BV-functions, which is a substitute of the Sobolev space of the marginal case.

In our setting of the maximizing problem, we suffer from the non-compactness due to the vanishing phenomenon and the non-reflexivity of the space of BV-functions. In order to overcome these difficulties, we use the fact that the family of maximizers of the Sobolev embedding with BV-functions is the set of characteristic functions on balls. Simultaneously, we give a characterization of maximizers of our problem to prove that the maximizers must form characteristic functions on balls and specify their radii and heights exactly.

This is a joint work with Prof. Michinori Ishiwata (Osaka University).