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The classical Gaffney inequality is a generalization
to differential forms of the Gaffney-Friedrichs inequality estimating the $L^2$ norm of the gradient of a vector
field that has either vanishing normal or tangential component on the boundary by the
$L^2$ norms of the vector field and its curl and divergence. For any open, bounded, smooth domain $\Omega \subset \mathbb{R}^{3},$ the Gaffney-Friedrichs
inequality reads
\begin{align*}
\left\lVert \nabla u \right\rVert^{2}_{L^{2}} \leq c \left( \left\lVert \operatorname*{curl}u \right\rVert^{2}_{L^{2}}
+ \left\lVert \operatorname*{div} u \right\rVert^{2}_{L^{2}}
+ \left\lVert u \right\rVert^{2}_{L^{2}} \right), \
\end{align*}
for all $ u \in W^{1,2}\left( \Omega; \mathbb{R}^{3} \right)$ with either $ \nu\times u =0 \text{ or } \nu\cdot u =0
\text{ on } \partial\Omega.$ For a particular boundary condition, the smallest admissible constant $c >0$ in the inequality is a
property of the domain and is greater than or equal to $1$ for any domain. In this talk I shall present a geometric characterization of all smooth domains for which the constant is $1.$ Some related results and extensions to the class of piecewise smooth domains
will also be discussed. This is a joint work with G.Csato and B. Dacorogna. |