Improved Sobolev's inequalities, relative entropy and fast diffusion equations

The difference of the two terms in Sobolev's inequality (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. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which also applies to other interpolation inequalities of Gagliardo-Nirenberg-Sobolev type.ou