Abstract

Quantitative stability results are known for more than 30 years, from the celebrated paper of G. Bianchi and H. Egnell. However, stability results with explicit, dimensionally sharp constants and optimal norms for the Sobolev inequality were obtained only recently, in a paper written in collaboration with M.J. Esteban, A. Figalli, R.L. Frank, and M. Loss. This has interesting consequences for the Gaussian logarithmic Sobolev inequality as well. The stability for the logarithmic Sobolev inequality can indeed be obtained either as a byproduct of the stability for the Sobolev inequality or by a direct proof. In a collaboration with G. Brigati and N. Simonov, results in stronger norms were also achieved under appropriate constraints. These progresses raise various new questions.

Abstract

The liquid drop model was originally introduced by Gamow in 1928 to model atomic nuclei. The model describes the competition between surface tension (which keeps the nuclei together) and Coulomb force (which corresponds to repulsion among the protons). Equilibrium shapes correspond to sets in the 3-dimensional Euclidean space which satisfy an equation that links the mean curvature of the boundary of the set to the Newtonian potential of the set. 

In this talk I will present the construction of toroidal surfaces, modelled on a family of Delaunay surfaces, with large volume which provide new equilibrium shapes for the liquid drop model. This work is in collaboration with M. del Pino and A. Zuniga.

Abstract

We consider a nonlinear nonlocal coupled system of beam-wave equations, modelling the dynamics of a degenerate plate and the behavior of suspension bridges. Since the damping of the system is "degenerate", this leads to a partially dissipative dynamical system for which unpredictable behaviors may appear. This is joint work with Maurizio Garrione and Emanuele Pastorino.

Abstract

Suppose that two nonlocal minimal surfaces are included one into the other and touch at a point. Then, they must coincide. But this is perhaps less obvious than what it seems at first glance.