# Jean Dolbeault

## CEREMADE, UniversitĂ© Paris-Dauphine

### Stability in Gagliardo-Nirenberg inequalities

Optimal constants and optimal functions are known in some functional inequalities. The next question is the stability issue: is the difference of the two terms controlling a distance to the set of optimal functions? A famous example is the case of Sobolevâ€™s inequalities: in 1991, Bianchi and Egnell proved that the difference of the two terms (with the optimal constant taken into account) is bounded from below by the \(H^1_0\) distance to the manifold of the Aubin-Talenti functions. They argued by contradiction and gave a very elegant although not constructive proof. Since then, estimating the new stability constant has been a challenge. In this lecture, I will address some results concerning subcritical Gagliardo-Nirenberg inequalities for which explicit constants can be provided. In the Euclidean space, the results are based on a joint work with Matteo Bonforte, Bruno Nazaret, and Nikita Simonov.