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

Jean Dolbeault (1) , Marta Garcia-Huidobro (2) , Raul Manásevich (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Departamento de Matemáticas [Santiago de Chile]
3 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]

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.

Mots clés: Interpolation; Gagliardo-Nirenberg inequalities; bifurcation; branches of solutions; elliptic equations; p-Laplacian; entropy; Fisher information; carré du champ method; Poincaré inequality; rigidity; uniqueness; rescaling; period; nonlinear Keller-Lieb-Thirring energy estimates

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Marta Garcia-Huidobro, Raul Manásevich. Interpolation inequalities in W1,p(S1) and carré du champ methods. 2019. ⟨hal-02003141v1⟩