Analysis Seminar

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SPECIAL AFTERNOON SEMINAR: Approximation by Cahn-Hoffman facets and the crystalline mean curvature flow

Speaker: Yoshikazu Giga, Tokyo University

Location: Warren Weaver Hall 1302

Date: Thursday, November 2, 2017, 3:30 p.m.

Synopsis:

We are interested in approximation of a general compact set in an Euclidean space by nicer sets. In fact, we show that every compact set can be monotonically approximated by a set admitting a certain vector field called the Cahn-Hoffman vector field. Such a set is called a Cahn-Hoffman facet. If the divergence of the minimal Cahn-Hoffman vector field is constant such a set is often called a Cheeger set, which has been widely studied by B. Kawohl, V. Caselles and others. More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner.

It turns out that this approximation is useful to construct suitable test functions necessary to establish comparison principle for level-set crystalline mean curvature flow equations. As a consequence, we obtained the well-posedness in arbitrary dimension. For a total variation flow of non-divergence type such a comparison result has been established by a joint work with M.-H. Giga and N. Pozar (2014). This lecture is based on my joint work with Norbert Pozar of Kanazawa University.