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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 theasymptotic behaviour of the solutions to some minimum problems for integralfunctionals on $\Omega\setminus K_j$, with Neumann boundary conditions on$\partial(\Omega\setminus K_j)$. We prove that the limit of these solutions isa minimiser of the same functional on $\Omega\setminus M$ subjected to atransmission 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 ischaracterised, and the link between the measure $\mu$ and the sequence $(K_j)$is expressed by means of suitable local minimum problems.

Transmission conditions obtained by homogenisation