Abstract

This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carre 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 W-1,W-p(S-1) 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 carre 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 not equal 2.