/sci/ - Science & Math

Consider the domain in pic related and the following equations.
[eqn]
\begin{align}
\partial_t u - \nabla \cdot (\beta \nabla u) &= f &&\text{in } \Omega_1 \cup \Omega_2 \\
u &= g &&\text{on } \partial\Omega \\
\left[u\right] &= a &&\text{on } \Gamma_\text{I} \\
\left[\beta \nabla u \cdot \mathbf{n} \right] &= b &&\text{on } \Gamma_\text{I}
\end{align}
[/eqn]
where [eqn]\left[v\right](\mathbf{\gamma}) = \left(\lim_{\mathbf{x}\to\mathbf{\gamma} \land \mathbf{x} \in \Omega_1} v(\mathbf{x}) \right)- \left(\lim_{\mathbf{x}\to\mathbf{\gamma} \land \mathbf{x} \in \Omega_2} v(\mathbf{x})\right)
[/eqn]
Basically this models functions with discontinuities along some irregular interface that partitions the domain. This is known as a parabolic interface problem if you want to look into some of the literature.

I'm trying to use this to model passive and active diffusion across a cellular membrane but can't figure out what kind of conditions to impose on [math]a[/math] and [math]b[/math] to get it to feel natural. All the papers I've looked into kind of decouple the interface conditions from the rest of the system too much (e.g. https://doi.org/10.1002/cnm.1132 ). I'd like to get things like Nernst potentials to show up properly. I know this is a niche topic but any ideas on how to proceed?