Salve a tutti, vorrei ricordarvi domani un doppio seminario; alle 17.00 in aula seminari, Eleonora Cinti presenterà "An improved quantitative fractional isoperimetric inequality" alle 17.45, stessa aula, Matteo Talluri presenterà "On a fractional Lin–Ni-Takagi type problem with Neumann boundary conditions" Di seguito gli abstract Siete tutti invitati a partecipare A presto e grazie per l'attenzione Marco An improved quantitative fractional isoperimetric inequality We present an improved version of the quantitative fractional isoperimetric inequality, in which a stronger notion of asymmetry appears. In particular, we show that the square root of the isoperimetric deficit controls, not only the Fraenkel asymmetry, but also a sort of "oscillation of the boundary". In the classical local setting the analogue result was obtained by Fusco and Julin. This is a joint project with E. M. Merlino and B. Ruffini. On a fractional Lin–Ni-Takagi type problem with Neumann boundary conditions We explore a fractional version of a semilinear Neumann problem, introduced by Lin–Ni and Takagi in the ’80s, that arises from the steady states of the Keller–Segel model with non–local diffusion. We study the system under two different “non–local Neumann conditions”: spectral and integral. While spectral conditions admit a Caffarelli-Silvestre type extension theorem, integral conditions do not. This absence makes it more difficult to obtain Liouville–type theorems and to show the non existence of non–trivial positive solutions. Based on a joint project with Eleonora Cinti.