Seminar Analysis: Giorgio Stefani (Scuola Normale Superiore di Pisa)

A distributional approach to fractional Sobolev spaces and fractional variation

Somewhat surprisingly, the first appearance of the concept of a fractional derivative is found in a letter written to de l'Hôpital by Leibniz in 1695. Since then, Fractional Calculus has fascinated generations of mathematicians and several definitions of fractional derivatives have appeared. In more recent years, the fractional operator defined as the gradient of the Riesz potential has received particular attention, since it has revealed to be a useful tool for the study of fractional-order PDEs and fractional Sobolev spaces. In a joint work with G. E. Comi, combining the PDE approach developed by Spector and his collaborators with the distributional point of view adopted by Šilhavý, we introduced new notions of fractional variation and fractional Caccioppoli perimeter in analogy with the classical BV theory. Within this framework, we were able to partially extend De Giorgi’s Blow-up Theorem to sets of locally finite fractional Caccioppoli perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets. In this talk, after a quick overview on Fractional Calculus, I will introduce the main features of the fractional operators involved and then give an account on the main results on the fractional variation we were able to achieve so far.

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