Geometric and functional inequalities play a crucial role in several problems arising in the calculus of variations, partial differential equations, geometry, etc. More recently, there has been a growing interest in the study of the stability for such inequalities. The basic question one wants to address is the following: suppose we have a functional inequality for which we know the sets of functions for which equality is attained. Can we prove, in some quantitative way, that if a function "almost attains the equality" then it is close (in some suitable sense) to a minimizer?
In recent years there have been several results in this direction, showing sharp stability results for the Euclidean/anisotropic isoperimetric inequality, the Brunn-Minkowski inequality on convex sets, the Gaussian isoperimetric inequality, Sobolev and Gagliardo-Nirenberg inequalities, etc. The aim of these lectures is to show different ways to attack these problems, together with some applications of these results.
More precisely, we first focused on the isoperimetric inequality showing three different approaches to prove a sharp stability result: via symmetrization techniques, via optimal transport theory, and via a second order stability analysis. Then we applied this result to two different variational problems: the first one concerns the shape of liquid drops/crystals under the action of an exterior potential force; the second one is the study of the isoperimetric problem on manifolds with density when the volume is small. We then focused on another family of inequalities: Sobolev and Gagliardo-Nirenberg. We discussed the stability issue for them, and as an application we obtained a quantitative rate of convergence to a stationary state for the critical mass Keller-Segel system.