Interface problem

Interface

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Assume we have $\phi_1$ and $\phi_2$,

$$\begin{align} (\beta \nabla p_1, \nabla v)_\Omega =& (f,v)_\Omega - <q_1,v>_\Gamma + (\beta \nabla \phi_1, \nabla v)_{\Omega^-}\\ (\beta \nabla p_2, \nabla v)_\Omega =& (f,v)_\Omega - <q_1,v>_\Gamma + (\beta \nabla \phi_2, \nabla v)_{\Omega^-}\\ \end{align}$$

Then we have:

$$\begin{align} u_1 = p_1 - \phi_1\\ u_2 = p_2 - \phi_2\\ \end{align}$$

Does $u_1 = u_2$?

$$\begin{align} (\beta \nabla (p_1 - p_2), \nabla v)_\Omega = (\beta \nabla (\phi_1-\phi_2), \nabla v)_{\Omega^-}\\ \end{align}$$

$$\begin{align} (\beta \nabla (p_1 - p_2), \nabla v)_{\Omega^+} + (\beta \nabla (p_1 - p_2), \nabla v)_{\Omega^-} = (\beta \nabla (\phi_1-\phi_2), \nabla v)_{\Omega^-}\\ \end{align}$$

$$\begin{align} (\beta \nabla (u_1 - u_2), \nabla v)_{\Omega^+} + (\beta \nabla (u_1 - u_2), \nabla v)_{\Omega^-} = 0\\ \end{align}$$

Noted $u_1 - u_2$ as $w$, then we have

$$\begin{align} (\beta \nabla w, \nabla v) =& 0\\ [w]_{\Gamma} =& w^+ - w^- = 0\\ [\beta w_n]_{\Gamma} =& \beta^+ w_n^+ - \beta^-w_n^- = 0?\\ w_{\partial \Omega} = 0 \end{align}$$