Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jean-Christophe Pain

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.

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

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 , 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.
Paper Structure (27 sections, 2 theorems, 90 equations)

This paper contains 27 sections, 2 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. A structural link between the Olech-Opial and Wirtinger inequalities
  3. The Wirtinger inequality
  4. The Olech-Opial inequality
  5. An integral identity
  6. Derivation of a weak Wirtinger inequality
  7. Comparison with the optimal constant
  8. Interpretation
  9. Application to Emden-Fowler type equations
  10. The Emden-Fowler equation
  11. Energy identity
  12. Control of the nonlinear term
  13. A priori estimate: a lower bound on the energy
  14. Role of the Olech-Opial inequality and interpretation
  15. A variational inequality linking Olech-Opial, Wirtinger and Emden-Fowler equations
  16. ...and 12 more sections

Key Result

Proposition 2.1

The constant $\frac{L}{2}$ in Eq. (two) is optimal.

Theorems & Definitions (3)

  • Proposition 2.1
  • proof
  • Theorem 4.1