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

Dolbeault, Jean; Esteban, Maria J.; Laptev, Ari; Loss, Michael (2014), One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: Remarks on duality and flows, Journal of The London Mathematical Society, 90, 2, p. 525-550. http://dx.doi.org/10.1112/jlms/jdu040

Type
Article accepté pour publication ou publié
Date
2014
Journal name
Journal of The London Mathematical Society
Volume
90
Number
2
Publisher
C. F. Hodgson
Pages
525-550
Publication identifier
http://dx.doi.org/10.1112/jlms/jdu040
Metadata
Show full item record
Author(s)
Dolbeault, Jean cc
Esteban, Maria J. cc
Laptev, Ari
Loss, Michael
Abstract (EN)
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 nonlinear 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 nonlinear 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 nonlinear elliptic equations and shows how eigenvalues of a linearized problem enter in 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 p<2. The algebraic part in the computations is very similar in both cases, although the case p<2 is definitely less standard.
Subjects / Keywords
Emden-Fowler transformation; Gagliardo-Nirenberg-Sobolev inequalities; continuity equation; stereographic projection; duality; action functional; second moment; sharp rates; gradient flow; distance on measure spaces; optimal transport; heat equation; interpolation; optimal constants; fast diffusion equation; Barenblatt solutions

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