Improved interpolation inequalities, relative entropy and fast diffusion equations

Understand

We consider a family of Gagliardo-Nirenberg-Sobolev interpolation inequalities which interpolate between Sobolev's inequality and the logarithmic Sobolev inequality, with optimal constants.

• The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions.
• We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants.
• Our approach is based on nonlinear evolution equations and improved entropy - entropy production estimates along the associated flow.