On the best constant in {G}affney inequality

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 $\nu \omega =0$ on $\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$