The infimum of the $L^{2}$ norm on the closed $2$-forms in a cohomology class defines a norm on $H^{2}(Y_{\epsilon},\mbox{${\mathbb{R}}$})$.

Define ${\mathcal{H}}^{1}$ to be the set of 1-forms $\alpha$ on $\overline{Y_{\epsilon}}$ with $d\alpha=0,d^{*}\alpha=0$ and with $(\alpha,\underline{\nu})=0$ on the boundary. Then the natural map from ${\mathcal{H}}^{1}$ to $H^{1}(Y_{\epsilon},\mbox{${\mathbb{R}}$})$ is an isomorphism.

If $F$ is any exact $2$-form on $\overline{Y_{\epsilon}}$ there is a unique $1$-form $\alpha$ such that $d^{*}\alpha=0,d\alpha=F,(\alpha,\underline{\nu})=0$ and $\alpha$ is $L^{2}$-orthogonal to ${\mathcal{H}}^{1}$. We have, for some fixed constant $C_{8}$, $\|\alpha\|_{L^{p}_{1}}\leq C_{8}\|F\|_{L^{p}}$.