Let $D \subset \mathbb{R}^n$, $(n \geq 2)$ be a bounded Lipschitz domain with connected boundary $\Gamma$. Let $D$ be an open subset relative to $\Gamma$. Given a function ${\bf f} \in H^{\frac{1}{2}}(D)$ (i.e. the $L^2-$based Bessel-potential space defined by the restriction from $\Gamma$ to $D$), does there exist an extension ${\bf F} \in H^{\frac{1}{2}}(\Gamma)$ to the whole boundary of ${\bf f}$, i.e., ${\bf F}|_D = {\bf f}$, which satisfies the condition $$ \int_\Gamma {\bf F} \nu d\sigma = 0 $$ where $\nu$ is the outward unit normal to $\Gamma$ and $d\sigma$ the surface measure which exist a.e. on $\Gamma$.
In other words, I am looking for a function ${\bf F} \in H^{\frac{1}{2}}(\Gamma)$ on the boundary $\Gamma$ such that the flux that passes through it is zero, i.e., the flux through $D$ should be equal to minus the flux that passes through $\Gamma \backslash \overline{D}$.
Does anyone know a theorem that guarantees such an extension, for some plausible conditions on the boundary? Or is it possible to construct such a function?