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arXiv · 2603.23239

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

Jean-Christophe Pain
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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.

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Jean-Christophe Pain. 2026-03-24. A Variational link between the Olech-Opial inequality, the Wirtinger inequality, and Emden-Fowler equations. https://arxiv.org/abs/2603.23239

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