One-dimensional Gagliardo Nirenberg Sobolev inequalities: remarks on duality and flows
File(s)
Accepted version
Author(s)
Dolbeault, J
Esteban, MJ
Laptev, A
Loss, M
Type
Journal Article
Abstract
his 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>2p>2 or to 1<p<21<p<2. The algebraic part in the computations is very similar in both cases, although the case 1<p<21<p<2 is definitely less standard.
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>2p>2 or to 1<p<21<p<2. The algebraic part in the computations is very similar in both cases, although the case 1<p<21<p<2 is definitely less standard.
Date Issued
2014-08-06
Date Acceptance
2014-03-08
Citation
Journal of the London Mathematical Society-Second Series, 2014, 90, pp.525-550
ISSN
1469-7750
Publisher
London Mathematical Society
Start Page
525
End Page
550
Journal / Book Title
Journal of the London Mathematical Society-Second Series
Volume
90
Copyright Statement
This is a pre-copyedited, author-produced PDF of an article accepted for publication in J. London Math. Soc. following peer review. The version of record Jean Dolbeault, Maria J. Esteban, Ari Laptev, and Michael Loss
One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: remarks on duality and flows
J. London Math. Soc. (2014) 90 (2): 525-550 first published online August 6, 2014 doi:10.1112/jlms/jdu040 is available online at: https://dx.doi.org/10.1112/jlms/jdu040
One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: remarks on duality and flows
J. London Math. Soc. (2014) 90 (2): 525-550 first published online August 6, 2014 doi:10.1112/jlms/jdu040 is available online at: https://dx.doi.org/10.1112/jlms/jdu040
Subjects
Science & Technology
Physical Sciences
Mathematics
CONCENTRATION-COMPACTNESS PRINCIPLE
STRONG MAXIMUM PRINCIPLE
FAST DIFFUSION EQUATION
ELLIPTIC-EQUATIONS
TIME ASYMPTOTICS
SELF-SIMILARITY
MASS-TRANSPORT
SHARP SOBOLEV
CONTINUITY
MANIFOLDS
Publication Status
Published