A Variational link between the Olech-Opial inequality, the Wirtinger inequality, and Emden-Fowler equations

TL;DR

This paper reveals a deep connection between classical inequalities and nonlinear differential equations, deriving optimal constants via variational methods linked to Emden-Fowler equations, bridging inequalities and boundary value problems.

Contribution

It introduces a novel variational framework connecting the Olech-Opial and Wirtinger inequalities through Emden-Fowler equations, providing explicit optimal constants.

Findings

Derived a nonlinear interpolation inequality linking the inequalities.

Characterized the optimal constant via a variational problem.

Obtained explicit expression of the constant using the Beta function.

Abstract

We establish a structural connection between the classical Olech-Opial inequality and the Wirtinger inequality. Using an integral identity involving the mixed energy term $uu'$, we derive a nonlinear interpolation inequality linking these two results. The optimal constant is characterized by a variational problem whose extremals satisfy an Emden-Fowler equation. An explicit expression of the optimal constant is obtained in terms of the Beta function. This approach provides a natural bridge between mixed-energy integral inequalities, classical spectral estimates, and nonlinear boundary value problems.