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--- abstract: '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.' address: - 'Scuola Internazionale Superiore di Studi Avanzanti, Trieste, Italy' - 'Istituto Nazionale di Alta Matematica, Unità di Ricerca di Firenze - DiMaI “U. Dini”, Università di Firenze, Italy' - 'Dipartimento di Matematica “G. Peano”, Università di Torino, Italy' author: - Gianni Dal Maso - Giovanni Franzina - Davide Zucco title: Transmission conditions obtained by homogenisation --- [**Keywords.** $\Gamma$-convergence, capacitary measures, Neumann sieve.]{} Introduction ============ This paper studies the asymptotic behaviour of the solutions $u_j$ to the equations $$-\Delta u_j +u_j=h \qquad \text{in $\Omega\setminus K_j$,}$$ with homogeneous Neumann boundary conditions on $\partial (\Omega\setminus K_j)$. Here and henceforth $\Omega$ is a bounded open set in $\mathbb{R}^n$, $n\ge 2$, and $(K_j)$ is a sequence of compact sets in $\mathbb{R}^n$. We assume that there exists a compact $(n-1)$-dimensional $C^1$ manifold $M$ with boundary, contained in $\Omega$, such that $$\label{eq:1.1} K_j \subset \big\{ x\in\mathbb{R}^n\,\,\colon\,\, {\rm dist}(x,M)\le\rho_j\big\}\,,$$ for a suitable sequence of positive numbers $(\rho_j)$ with $\rho_j\to0^+$ as $j\to\infty$. It is well known that $u_j$ is the minimiser of the functional $$\int_{\Omega\setminus K_j}|\nabla u|^2dx +\int_{\Omega\setminus K_j} u^2dx - 2\int_{\Omega\setminus K_j}hu\, dx\,,$$ and this property will be the starting point of our analysis. More in general, in this paper we will consider sequences of minimisers of functionals of the form $$\label{fct0} \int_{\Omega{\setminus}K_j} f(\nabla u)\,dx +\int_{\Omega\setminus K_j} g(x,u)\,dx\, ,$$ where $f$ and $g$ are suitable functions satisfying standard convexity and growth conditions (see Section \[subsec:2.1\] for details). A significant instance included in our analysis will be that of the functions $f(\nabla u)=|\nabla u|^p$, with $1<p\le n$, and $g(x,u)=|u-h(x)|^q$, with $1\le q<\infty$ and $h\in L^q(\Omega)$. This kind of question is related with the so-called *Neumann sieve problem* that was proposed by Sanchez and Palencia, who gave in [@SP2] a formal asymptotic expansion of the solution. It was then studied by Attouch, Damlamian, Murat, and Picard (see [@AP; @D; @M; @P]) in the case where the perforations are composed of open balls periodically distribuited over the manifold. For related studies on the asymptotic behaviour of periodically-perforated domains see [@AB; @AB2; @C1; @C2; @C3; @DV; @SP1; @SP3]. A slight modification (see Section \[sec:3\]) of the results of [@Cor] shows that, under our general conditions, there exists a subsequence of $(K_j)$ (not relabelled) satisfying such that the minimisers $u_j$ of converge in $L^q(\Omega)$ as $j\to\infty$ to the minimiser of a functional of