  This paper is devoted to an extension of rigidity results for nonlinear
differential equations, based on carr{\'e} 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 $\ge$
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 carr{\'e} 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$\ne$2.
