  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 $\partial\Omega.$
