On the best constant in {G}affney inequality

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 $\partial\Omega.$