  Given a bounded open set in $\mathbb{R}^n$, $n\ge 2$, and a sequence $(K_j)$
of compact sets converging to an $(n-1)$-dimensional manifold $M$, we study the
asymptotic behaviour of the solutions to some minimum problems for integral
functionals on $\Omega\setminus K_j$, with Neumann boundary conditions on
$\partial(\Omega\setminus K_j)$. We prove that the limit of these solutions is
a minimiser of the same functional on $\Omega\setminus M$ subjected to a
transmission condition on $M$, which can be expressed through a measure $\mu$
supported on $M$. The class of all measures that can be obtained in this way is
characterised, and the link between the measure $\mu$ and the sequence $(K_j)$
is expressed by means of suitable local minimum problems.
