On the best constant in Gaffney inequality (Q1677456)

Let \(\Omega\subset{\mathbb R}^n\) be a bounded open set with smooth boundary, and let \(\nu\) be the outward unit vector field normal to \(\partial\Omega\), which may be considered as a \(1\)-differential form by using the scalar product. Let \(W^{1,2}_T(\Omega;\Lambda^k)\) (respectively, \(W^{1,2}_N(\Omega;\Lambda^k)\)) be the space of \(k\)-differential forms \(\omega\) in the Sobolev space \(W^{1,2}(\Omega;\Lambda^k)\) with boundary condition \(\nu\wedge\omega=0\) (respectively, \(\nu\lrcorner\,\omega=0\)). The Gaffney inequality states that there is some \(C>0\) such that \[ \|\nabla\omega\|_{L^2}^2\leq C(\|d\omega\|_{L^2}^2+\|\delta\omega\|_{L^2}^2+\|\omega\|_{L^2}^2) \] for all \(\omega\in W^{1,2}_T(\Omega;\Lambda^k)\cup W^{1,2}_N(\Omega;\Lambda^k)\). There are generalizations of this inequality to classes of non-smooth domains. Denoting the optimal constants by \(C=C_T(\Omega,k)>0\) (respectively, \(C=C_N(\Omega,k)>0\)), it is first shown that \(C_T(\Omega,k),C_N(\Omega,k)\geq1\). For \(k=0,n\), this is clearly an equality. For degrees \(1\leq k\leq n-1\), the main theorem of the paper states that \(C_T(\Omega,k)=1\) (respectively, \(C_N(\Omega,k)=1\)) is achieved if and only if \(\Omega\) is \((n-k)\)-convex (respectively, \(k\)-convex). Here, \(k\)-convexity means that the principal curvatures \(\gamma_1,\dots,\gamma_{n-1}\) of \(\partial\Omega\) satisfy \(\gamma_{i_1}+\dots+\gamma_{i_k}\geq0\) for all indices \(1\leq i_1<\dots\leq i_k< n-1\); in particular, \(1\)-convexity is the usual convexity. Other characterizations of this property are also given. The smoothness of \(\Omega\) is essential in this theorem; indeed, it is also shown that \(C_T(\Omega,k),C_N(\Omega,k)=1\) if \(\Omega\) is any (possibly non-convex) polytope.