Seminar Analysis and Mathematical Physics: Mattia Freguglia (Scuola Normale Superiore di Pisa)

A nonlocal approximation of the area functional in codimension two

In this seminar, we will introduce, for any $s\in (0,1)$, a notion of s-fractional volume on closed, oriented, codimension-two surfaces in Euclidean space. We will then show that this quantity Gamma-converges, with respect to the flat distance, and converges pointwise to the codimension-two Hausdorff measure, as s goes to 1. In particular, we will see how this fractional volume can be interpreted as the codimension-two analogue of the well-established notion of fractional perimeter. If time permits, we will discuss a possible generalization of this definition to higher codimension, along with some (equi)compactness properties related to these fractional volumes. Based on a joint project with Michele Caselli and Nicola Picenni.

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