Minimisers of a fractional seminorm and nonlocal minimal surfaces

12.06.2024 14:00 - 14:45

Luca Lombardini (TU Wien)


The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space.
In this seminar, I will relate these two types of variational problems. Specifically, I will show the connection between the nonlocal minimal surfaces and the minimisers of the -seminorm. In particular, I will prove that a function is a minimiser for the fractional seminorm if and only if its level sets are minimisers for the fractional perimeter. I will also provide an existence result for minimisers and study their qualitative and regularity properties.
The topics that I will present come from a series of papers written together with Claudia Bucur, Serena Dipierro, José Mazon and Enrico Valdinoci.

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HS 2, EG, OMP 1