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

Dolbeault, Jean; Garcia-Huidobro, Marta; Manásevich, Raul (2020), Interpolation inequalities in W1,p(S1) and carré du champ methods, Discrete and Continuous Dynamical Systems. Series A, 40, 1, p. 375-394. 10.3934/dcds.2020014

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Type
Article accepté pour publication ou publié
Date
2020
Journal name
Discrete and Continuous Dynamical Systems. Series A
Volume
40
Number
1
Publisher
AIMS - American Institute of Mathematical Sciences
Pages
375-394
Publication identifier
10.3934/dcds.2020014
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
rescaling; period; nonlinear Keller-Lieb-Thirring energy estimates; bifurcation; carré du champ method; p-Laplacian; Fisher information; entropy; elliptic equations; Interpolation; Gagliardo-Nirenberg inequalities; rigidity; Poincaré inequality; uniqueness; branches of solutions

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