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Le problème de la passoire de Neumann. (The Neumann sieve problem). (French) Zbl 0629.35031

Let \(N\in {\mathbb{N}}\), \(N\geq 3\), \(O_{\epsilon}\subset \{x\in {\mathbb{R}}^ N:\) \(x_ i\in]0,1[\), \(i=1,...,N-1\), \(x_ N=0\}\) where \(O_{\epsilon}=\epsilon^{1/(N-2)_ O}\) and O is an open subset in \(\Sigma \equiv \{x\in {\mathbb{R}}^ N:\) \(x_ N=0\}\), \(T_{\epsilon}=\cup \{\epsilon (O_{\epsilon}+\gamma):\gamma_ i\in {\mathbb{Z}}\), \(i=1,...,N- 1,\gamma_ N=0\}\), \(\Omega\) an open regular subset of \({\mathbb{R}}^ N\) such that \(\Omega^{\pm}\equiv \{x\in \Omega:\) \(x_ N^{\pm}>0\}\) are not empty open subsets with regular boundary with the exception of points of \(\partial \Omega \cap \Sigma\) where we suppose that \(\Sigma\) is transversal and moreover \(T_{\epsilon}\cap \Omega =\emptyset\). Let \(V_ 0\) be a subspace of \(H^ 1(\Omega^+)\times H^ 1(\Omega^-)\) of the functions which satisfy a variational condition on \(\partial \Omega \setminus \Sigma\), \(\gamma_{\pm}u\) the trace of \(u\in H^ 1(\Omega^{\pm})\) on \(\partial \Omega^{\pm}\cap \Sigma\) and \([u]=\gamma_+u_ 1=\gamma_-u_ 2\) if \(u\equiv (u_ 1,u_ 2)\in V_ 0.\)
Theorem 1. Let \(f\in V_ 0'\). If \(u_{\epsilon}\in V_ 0\) verify \([u_{\epsilon}]=0\) on \(T_{\epsilon}\) and \[ \int_{\Omega^+\cup \Omega^-} (\nabla u_{\epsilon}\cdot \nabla v+u_{\epsilon}v)dx=<f,v> \] for all \(v\in V_ 0\) such that \([v]=0\) on \(T_{\epsilon}\), then \(u_{\epsilon}\rightharpoonup u\in V_ 0\) and there exists \(c>0\) such that \[ \int_{\Omega^+\cup \Omega^-} (\nabla u\nabla v+uv)dx+c\int_{\Sigma} [u][v] d\sigma =<f,v> \text{ for all } v\in V_ 0. \] Theorem 2. Let \(f\in V_ 0'\). If \(u_{\epsilon}\in V_ 0\) verifies \([u_{\epsilon}]\geq 0\) on \(T_{\epsilon}\) and \[ \int_{\Omega^+\cup \Omega^-} (<\nabla u_{\epsilon},\nabla v- \nabla u_{\epsilon}>+<u_{\epsilon},v-u_{\epsilon}>)dx\geq <f,v- u_{\epsilon}> \] for each \(v\in V_ 0\) such that \([v]\geq 0\) on \(T_{\epsilon}\), then \(u_{\epsilon}\rightharpoonup u\in V_ 0\) and there exists \(c>0\) such that \[ \int^{0}_{\Omega^+\cup \Omega^-} (\nabla u,\nabla v-\nabla u>+<u,v-u>)dx+c\int_{\Sigma}[u]^-([v]-[u]) d\sigma \geq <f,v-u>\text{ for all } v\in V_ 0. \] Some extensions are also given.
Reviewer: G.Bottaro

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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