On the best constant in {G}affney inequality

TL;DR

This paper investigates the optimal constant in Gaffney inequality under specific boundary conditions, providing insights into its precise value and implications for differential forms in mathematical analysis.

Contribution

It determines the best constant in Gaffney inequality for certain boundary conditions, advancing understanding of inequalities involving differential forms.

Findings

Identifies the optimal constant in Gaffney inequality for specified boundary conditions.

Provides bounds and conditions under which the best constant is achieved.

Enhances theoretical understanding of differential form inequalities in mathematical analysis.

Abstract

We discuss the value of the best constant in Gaffney inequality namely $$ \lVert \nabla \omega \rVert_{L^{2}}^{2}\leq C\left( \lVert d\omega\rVert_{L^{2}}^{2}+\lVert \delta\omega\rVert_{L^{2}% }^{2}+\lVert \omega\rVert_{L^{2}}^{2}\right) $$ when either $\nu\wedge\omega=0$ or $\nu\,\lrcorner\,\omega=0$ on $\partial\Omega.$