Fick’s law selects the Neumann boundary condition

We show that the Neumann boundary condition appears along the boundary of an inner domain when the diffusivity of the outer domain goes to zero. We take Fick’s diffusion law with a bistable reaction function, and the diffusivity is 1 in the inner domain and $ϵ>0$ in the outer domain. The convergence of the solution as $ϵ\to 0$ is shown, where the limit satisfies the Neumann boundary condition along the boundary of an inner domain. This observation says that the Neumann boundary condition is a natural choice of boundary conditions when Fick’s diffusion law is taken.