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Interpolation inequalities in W1,p(S1) and carré du champ methods

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DoGHMa-2019.pdf (555.5Kb)
Date
2019
Publisher city
Paris
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Publishing date
02-2019
Collection title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Link to item file
https://hal.archives-ouvertes.fr/hal-02003141
Dewey
Probabilités et mathématiques appliquées
Sujet
nonlinear Keller-Lieb-Thirring energy estimates; period; rescaling; uniqueness; Poincaré inequality; rigidity; Interpolation; Gagliardo-Nirenberg inequalities; bifurcation; branches of solutions; elliptic equations; p-Laplacian; entropy; Fisher information; carré du champ method
URI
https://basepub.dauphine.fr/handle/123456789/18553
Collections
  • CEREMADE : Publications
Metadata
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Author
Dolbeault, Jean
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Garcia-Huidobro, Marta
status unknown
Manásevich, Raul
63 Centre de Modélisation Mathématique / Centro de Modelamiento Matemático [CMM]
18014 Departamento de Ingeniería Matemática [Santiago] [DIM]
Type
Document de travail / Working paper
Item number of pages
19
Abstract (EN)
This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carré du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in W1,p(S1) with p ≥ 2. Mostly for numerical reasons, we relate our estimates with issues concerning periodic dynamical systems. Our interpolation inequalities have a dual formulation in terms of generalized spectral estimates of Keller-Lieb-Thirring type, where the differential operator is now a p-Laplacian type operator. It is remarkable that the carré du champ method adapts to such a nonlinear framework, but significant changes have to be done and, for instance, the underlying parabolic equation has a nonlocal term whenever p≠2.

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