One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows

Abstract

This paper is devoted to one-dimensional interpolation Gagliardo–Nirenberg–Sobolev inequalities. We study how various notions of duality, transport and monotonicity of functionals along flows defined by some non-linear diffusion equations apply. We start by reducing the inequality to a much simpler dual variational problem using mass transportation theory. Our second main result is devoted to the construction of a Lyapunov functional associated with a non-linear diffusion equation, that provides an alternative proof of the inequality. The key observation is that the inequality on the line is equivalent to Sobolev's inequality on the sphere, at least when the dimension is an integer, or to the critical interpolation inequality for the ultraspherical operator in the general case. The time derivative of the functional along the flow is itself very interesting. It explains the machinery of some rigidity estimates for non-linear elliptic equations and shows how eigenvalues of a linearized problem enter into the computations. Notions of gradient flows are then discussed for various notions of distances. Throughout this paper, we shall deal with two classes of inequalities corresponding either to p > 2 or to 1 < p < 2 . The algebraic part in the computations is very similar in both cases, although the case 1 < p < 2 is definitely less standard.