Interpolation inequalities in <inline-formula><tex-math id="M1">\begin{document}$ \mathrm W^{1,p}( {\mathbb S}^1) $\end{document}</tex-math></inline-formula> and <i>carré du champ</i> methods

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 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 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≠2.