1. One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows
- Author
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Jean Dolbeault, Michael Loss, Maria J. Esteban, Ari Laptev, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Department of Mathematics [Imperial College London], Imperial College London, School of Mathematics - Georgia Institute of Technology, Georgia Institute of Technology [Atlanta], and NSF grant: DMS-1301555
- Subjects
Pure mathematics ,CONCENTRATION-COMPACTNESS PRINCIPLE ,General Mathematics ,CONTINUITY ,Duality (optimization) ,Monotonic function ,Gagliardo-Nirenberg-Sobolev inequalities ,01 natural sciences ,TIME ASYMPTOTICS ,0101 Pure Mathematics ,Sobolev inequality ,gradient flow ,distance on measure spaces ,26D10 ,46E35 ,35K55 ,FAST DIFFUSION EQUATION ,Mathematics - Analysis of PDEs ,action functional ,stereographic projection ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,STRONG MAXIMUM PRINCIPLE ,Eigenvalues and eigenvectors ,Mathematics ,Science & Technology ,heat equation ,continuity equation ,010102 general mathematics ,ELLIPTIC-EQUATIONS ,optimal constants ,Interpolation inequality ,interpolation ,second moment ,010101 applied mathematics ,Sobolev space ,sharp rates ,Nonlinear system ,Barenblatt solutions ,optimal transport ,Flow (mathematics) ,Emden-Fowler transformation ,Physical Sciences ,MANIFOLDS ,MASS-TRANSPORT ,duality ,SELF-SIMILARITY ,SHARP SOBOLEV ,Analysis of PDEs (math.AP) - 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 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
- Published
- 2014