Convergence to equilibrium via quantitative stability

Geometric and functional inequalities play a crucial role in several PDE problems.

Very recently there has been a growing interest in studying the stability for such inequalities. The basic question one wants to address is the following:

Suppose we are given a functional inequality for which minimizers are known. Can we prove, in some quantitative way, that if a function “almost attains the equality” then it is close to one of the minimizers?

Actually, in view of applications to PDEs, a even more general and natural question is the following: suppose that a function almost solve the Euler-Lagrange equation associated to some functional inequality. Is this function close to one one of the minimizers?

While in the first case the answer is usually positive, in the second case one has to face the presence of bubbling phenomena.

In this talk I’ll give a overview of these general questions using some concrete examples, and then present recent applications to some fast diffusion equation related to the Yamabe flow.