Norms in $H^{-1/2}$, an inequality.

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Let

$\|\mu\|_{-1/2,T_h}:=\inf\{\|\sigma\|_{div,T_h}:\sigma\in H(div,\Omega),\,\mu=\sigma n^K,\textrm{on }\partial K,K\in T_h\}.$

$T_h$ is a partición of $\Omega$ (usual in finite elements), composed of triangles.

$n^K$ is the normal vector over an edge of the triangle $K\in T_h$, pointing outward of $K$. $H(div,\Omega)$ is the set of vectors with components in $L^2(\Omega)$ with divergence in $L^2(\Omega)$ and

$\|\sigma\|_{div,T_h}^2:=\displaystyle\sum_{K\in T_h}(\|\sigma\|_{L^2(K)^2}^2+\|div(\sigma)\|^2_{L^2(\Omega)})$.

It is true the following?

$\displaystyle\sum_{K\in T_h}\|\mu\|_{-1/2,K}^2\leq C \|\mu\|_{-1/2,T_h}^2$

Where $C$ is a constant not depending of the number of triangles of $T_h$.