AJS - Marco Picerni

Speaker: Marco Picerni (SISSA) Title: The Extended Maz'ya-Shaposhnikova Formula: From Mass at Infinity to Generalized Young Measures Abstract: The Maz'ya-Shaposhnikova formula shows that fractional Sobolev seminorms approximate the Lp norm as the fractional parameter approaches zero. However, this result requires the function to belong to a Lebesgue space, meaning it must decay at infinity. In this talk, I will analyze what happens when functions have non-vanishing mass at infinity, such as characteristic functions of unbounded sets. While the asymptotic behavior is known for fractional perimeters, the limit of localized Gagliardo seminorms for functions with non-vanishing mass at infinity remained unexplored until recently. For p=2 and functions admitting a radial limit, it has been shown that the limit procedure yields an interaction energy between the values of the function inside the domain and its limit values at infinity. I will present a framework based on the novel notion of projectability at infinity. By using Generalized Young Measures, we define the Maz'ya-Shaposhnikova trace of a function at infinity. This approach accounts for oscillations and concentration phenomena, allowing us to compute the limit of localized fractional seminorms for functions which may have irregular behaviors at infinity.