Interpolation inequalities in W1,p(S1) and carr{\'e} du champ methods

TL;DR

This paper extends rigidity results for nonlinear differential equations using carr{\'e} du champ methods, establishing explicit interpolation inequalities in W1,p(S1) with applications to spectral estimates and nonlinear operators.

Contribution

It introduces new explicit interpolation inequalities in W1,p(S1) for p ≥ 2, adapting carr{\'e} du champ methods to nonlinear p-Laplacian frameworks with nonlocal terms.

Findings

Derived explicit constants for interpolation inequalities in W1,p(S1).

Linked inequalities to spectral estimates of Keller-Lieb-Thirring type.

Adapted carr{\'e} du champ methods to nonlinear p-Laplacian operators.

Abstract

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.