  Using techniques of the theory of semigroups of linear operators we study the
question of approximating solutions to equations governing diffusion in thin
layers separated by a semi-permeable membrane. We show that as thickness of the
layers converges to $0$, the solutions, which by nature are functions of $3$
variables, gradually lose dependence on the vertical variable and thus may be
regarded as functions of $2$ variables. The limit equation describes diffusion
on the lower and upper sides of a two-dimensional surface (the membrane) with
jumps from one side to the other. The latter possibility is expressed as an
additional term in the generator of the limit semigroup, and this term is build
from permeability coefficients of the membrane featuring in the transmission
conditions of the approximating equations (i.e., in the description of the
domains of the generators of the approximating semigroups). We prove this
convergence result in the spaces of square integrable and continuous functions,
and study the way the choice of transmission conditions influences the limit.
