Existence of minimizers for the Bianchi-Egnell stability inequality

Abstract:
Associated to Sobolev's inequality, the stability inequality due to Bianchi and Egnell bounds the 'Sobolev deficit' of a function in terms of a constant $c_{BE} > 0$ times its squared $\dot{H}^1$-distance to the manifold of Sobolev optimizers.
In this talk, I will present some recent results concerning the Bianchi-Egnell inequality. Based on new strict upper bounds for the best constant $c_{BE}$, the existence of a minimizer for $c_{BE}$ can be proved by exploiting symmetry and convexity properties of the associated quotient functional. The same findings apply for the fractional Sobolev inequality of order $s \in (0, d/2)$, provided that the space dimension $d$ is at least two.

If $d =1$ on the other hand, the Bianchi-Egnell has a surprising different behavior, which I will also address in my talk.