The purpose of these pages is to collect a set of notes that are a result of several talks and
minicourses delivered here and there in the world (Milan, Cortona, Pisa, Roma, Santiago del …

The optimal Orlicz target space is exhibited for embeddings of fractional-order Orlicz–
Sobolev spaces in R n. An improved embedding with an Orlicz–Lorentz target space, which …

We establish the equivalence between the Sobolev seminorm‖∇ u‖ L p and a quantity
obtained when replacing strong L p by weak L p in the Gagliardo seminorm| u| W s, p …

We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit
constants. Moreover, the constants have the correct behavior in the limit of large dimensions …

We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching
Yau-type result: in every closed, $ n $-dimensional Riemannian manifold we construct …

The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-
phase free boundary problem. The problem of classifying stable (or minimal) homogeneous …

We show that on S1. 1= pd 2/Sd1. 1/the conformally invariant Sobolev inequality holds with
a remainder term that is the fourth power of the distance to the optimizers. The fourth power …

We consider a nonlocal isoperimetric problem defined in the whole space R^N, whose
nonlocal part is given by a Riesz potential with exponent α∈(0,N-1). We show that critical …

We establish quantitative properties of minimizers and stable sets for nonlocal interaction
functionals, including the $ s $-fractional perimeter as a particular case. On the one hand …

The quantitative isoperimetric inequality and related topics | Bulletin of Mathematical Sciences
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