  Motivated by biological applications on tumour invasion through thin
membranes, we study a porous-medium type equation where the density of the cell
population evolves under Darcy's law, assuming continuity of both the density
and flux velocity on the thin membrane which separates two domains. The
drastically different scales and mobility rates between the membrane and the
adjacent tissues lead to consider the limit as the thickness of the membrane
approaches zero. We are interested in recovering the effective interface
problem and the transmission conditions on the limiting zero-thickness surface,
formally derived by Chaplain et al. (2019), which are compatible with nonlinear
generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates
and compactness arguments as well as on the construction of a suitable
extension operator which allows to deal with the degeneracy of the mobility
rate in the membrane, as its thickness tends to zero.
