  Biological, physical, medical, and numerical applications involving membrane
problems on different scales are numerous. We propose an extension of the
standard Turing theory to the case of two domains separated by a permeable
membrane. To this aim, we study a reaction-diffusion system with zero-flux
boundary conditions on the external boundary and Kedem-Katchalsky membrane
conditions on the inner membrane. We use the same approach as in the classical
Turing analysis but applied to membrane operators. The introduction of a
diagonalization theory for compact and self-adjoint membrane operators is
needed. Here, Turing instability is proven with the addition of new
constraints, due to the presence of membrane permeability coefficients. We
perform an explicit one-dimensional analysis of the eigenvalue problem,
combined with numerical simulations, to validate the theoretical results.
Finally, we observe the formation of discontinuous patterns in a system which
combines diffusion and dissipative membrane conditions, varying both diffusion
and membrane permeability coefficients. The case of a fast reaction-diffusion
system is also considered.
