We suppose that $\mathcal{H}= L^2(\Omega)$, $\mathcal{\widetilde{H}}= L^2(\Omega\backslash\Gamma) $ and $ \mathcal{H}_1=H^1(\Omega)$, $\mathcal{\widetilde{H}}_1= H^1(\Omega\backslash\Gamma)$. It follows that $\mathcal{H}= \mathcal{\widetilde{H}}$ almost every where since $\Omega = \Omega \backslash\Gamma $ almost everywhere. We have linear operators
$J: \mathcal{H} \to \mathcal{\widetilde{H}}$, $J':\mathcal{\widetilde{H}} \to \mathcal{H} $
$J_1: \mathcal{H}_1 \to \mathcal{\widetilde{H}}_1$, $J'_1: \mathcal{\widetilde{H}}_1 \to\mathcal{H}_1$.
Let $J=J'=Identity$ and $J_1= Identity$.
We define the operator $J'_1: \mathcal{\widetilde{H}}_1 \to\mathcal{H}_1$ by
(J'_1f)(x,y) = {\begin{array}{*{20}{c}} {f(x,y), \quad if\quad (x,y) \in \Omega \backslash \Gamma }\\ {{\beta _0}\quad if \quad(x,y) \in \Gamma }. \end{array}}
where $\mathop {\lim f(x,y)}\limits_{(x,y) \to ({x_0},{y_0})} = {\beta _0}, \quad({x_0},{y_0}) \in \Gamma. $