I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your help would be greatly appreciated. I try to summarize the setting hereby:
Let there be a convex domain $\Omega \subset \mathbb{R}^2$ with two subdomains $\Omega_1, \Omega_2$ divided by the 1-dimensional path $\gamma$ (see picture). sketch of domain $\Omega$ with its subdomains
Given the Hilbert spaces \begin{align} L &= \{r = (r_1, r_2, r_{\tau}) \in L_2(\Omega_1) \times L_2(\Omega_2) \times L_2(\gamma)\}\\ W &= \{v = (v_1, v_2, v_f) \in H(div,\Omega_1) \times H(div,\Omega_2) \times H(div_\tau,\gamma): v_i \cdot n_i \in L_2(\gamma), i=1,2\} \end{align}
we let $r = (r_1, r_2, r_{\tau}) \in M$ and $\varphi = (\varphi_1, \varphi_2, \varphi_\gamma) \in H^2(\Omega_1) \times H^2(\Omega_2) \times H^2(\gamma)$ be the solution of the PDE
\begin{align} -\Delta{\varphi} = \tilde{r} \text{ on } \Omega\\ \varphi = 0 \text{ on } \Gamma \end{align} with $\tilde{r} \in L_2(\Omega)$ given by $\tilde{r}\vert_{\Omega_i} = r_i, \ i=1,2$ and \begin{align} -\Delta_\tau \varphi_\gamma = r_\tau \text{ on } \gamma\\ \varphi_\gamma = 0 \text{ on } \partial \gamma \end{align} We pose \begin{align} v_i &= -\nabla \varphi \vert_{\Omega_i}, \ i=1,2\\ v_\gamma &= -\nabla_\tau \varphi_\gamma \end{align} so $v=(v_1,v_2,v_\gamma) \in W$ and note that \begin{align} \text{div} (v_i) &= r_i, \ i=1,2\\ \text{div}_\tau (v_\gamma) &= r_\tau \end{align} $\Delta_\tau$ and $\nabla_\tau$ are the directional Laplacian and gradient along $\gamma$. Note that $v_1 \cdot n_1 = -v_2 \cdot n_2 \in L_2(\gamma)$ because $v_i \in (H^1(\Omega))^2$. Furthermore we define the following norms: \begin{align} ||v||_{0,\Omega_i}^2 = \langle v,v \rangle_{0,\Omega} = \int_{\Omega} v \cdot v \ d\omega\\ ||v||_{0,\gamma}^2 = \langle v,v \rangle_{0,\gamma} = \int_{\gamma} v \cdot v \ d \Gamma \end{align} with $u \cdot v$ is the dot product. The proposed inequality is \begin{multline} ||\tilde{r}||_{0,\Omega}^2 + ||r_\tau||_{0,\gamma}^2 + ||\nabla \varphi||_{0,\Omega}^2 + ||\nabla_\tau \varphi_\gamma||_{0,\gamma}^2 + 2||v_1 \cdot n_1 ||_{0,\gamma}^2\\ \leq (1+C(\Omega))||\tilde{r}||_{0,\Omega}^2+(1+C(\gamma))||r_\tau||_{0,\gamma}^2 + C(\Omega) ||\tilde{r}||_{0,\Omega}^2 \end{multline} with constants $C(\Omega), C(\gamma) > 0$.
The authorial intention I guessed:
Friedrichs inequality leads to \begin{align} ||\tilde{r}||_{0,\Omega}^2 + ||\nabla \varphi||_{0,\Omega}^2 \leq (1+C(\Omega))||\tilde{r}||_{0,\Omega}^2\\ ||r_\tau||_{0,\gamma}^2 + ||\nabla_\tau \varphi_\gamma||_{0,\Omega}^2 \leq (1+C(\gamma))||r_\tau||_{0,\gamma}^2\\ \end{align}
so it's sufficient to proof \begin{equation} 2 ||v_1 \cdot n_1||_{0,\gamma}^2 \leq C(\Omega) ||\tilde{r}||_{0,\Omega}^2 \end{equation} The only way I see to get an inequality of the norms $||\cdot||_{0,\gamma}$ and $||\cdot||_{0,\Omega}$ is Gauss' divergence theorem. Apparently it holds true that: \begin{equation} \int_{\gamma} |v_1 \cdot n_1| d\Gamma \leq \sum_{i=1}^2 \int_{\Omega_i} |\text{div} (v_i)| d\omega = \int_{\Omega} |\tilde{r}| d\omega \end{equation} because $\gamma \subset \partial \Omega_i,\ i=1,2$.
Unfortunately this is an inequality of $L_1$ norms that does NOT imply the same inequality for $L_2$ norms $||\cdot||_{0,\gamma}$ and $||\cdot||_{0,\Omega}$.
Source (section 4.3, p. 12 [Existence and uniqueness of the solution]):
Jérôme Jaffré, Vincent Martin, Jean Roberts. Modeling Fractures and Barriers as Interfaces for Flow in Porous Media. [Research Report] RR-4848, INRIA. 2003. https://hal.inria.fr/inria-00071735/document
