DERIVATION OF EFFECTIVE TRANSMISSION CONDITIONS FOR DOMAINS SEPARATED BY A MEMBRANE FOR DIFFERENT SCALING OF MEMBRANE DIFFUSIVITY

We consider a system of non-linear reaction diffusion equations in a domain consisting of two bulk regions separated by a thin layer with periodic structure. The thickness of the layer is of order is an element of, and the equations inside the layer depend on the parameter is an element of and an additional parameter gamma is an element of[-1, 1), which describes the size of the diffusion in the layer. We derive effective models for the limit is an element of -> 0, when the layer reduces to an interface Sigma between the two bulk domains. The effective solution is continuous across Sigma for all gamma is an element of [-1, 1). For gamma is an element of(-1, 1), the jump in the normal flux is given by a non-linear ordinary differential equation on Sigma. In the critical case gamma = -1, a dynamic transmission condition of Wentzell-type arises at the interface Sigma.