Abstract:
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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