Speaker
Jean Dolbeault
Abstract
Optimal functions and optimal constants are known in various functional inequalities, for instance in Sobolev’s inequality. In 1991, Bianchi and Egnell proved that the deficit in Sobolev’s inequality (with p=2, on the Euclidean space), that is, the difference of the two sides of the inequality written with the optimal constant, controls the distance to the manifold of the Aubin-Talenti optimal functions in a strong Sobolev norm. An issue with the method is that the new constant is obtained by compactness and nothing is known on its value. The purpose of this lecture is to review some examples of related functional inequalities in which one can at least give an estimate of the stability constant, with a special emphasis on critical inequalities. In the case of Sobolev’s inequality, a constructive result on the Bianchi-Egnell stability estimate will be presented, that has been obtained recently in collaboration with M.J. Esteban, A. Figalli, R. Frank and M. Loss.