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

Dolbeault, Jean; Garcia-Huidobro, Marta; Manásevich, Raul (2019), Interpolation inequalities in W1,p(S1) and carré du champ methods. https://basepub.dauphine.fr/handle/123456789/18553

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DoGHMa-2019.pdf (555.5Kb)
Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-02003141
Date
2019
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Published in
Paris
Pages
19
Metadata
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Author(s)
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Garcia-Huidobro, Marta

Manásevich, Raul
Centre de Modélisation Mathématique / Centro de Modelamiento Matemático [CMM]
Departamento de Ingeniería Matemática [Santiago] [DIM]
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.
Subjects / Keywords
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

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