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Homogenization of two heat conductors with an interfacial contact resistance. (English) Zbl 1083.35014

Consider a bounded open set \(\Omega \subset {\mathbb R}^n\), \(\Omega = \Omega_1^{\varepsilon} \cup {\overline {\Omega_2^{\varepsilon}}}\) with \(\Omega_i^{\varepsilon}\), open and with a periodic structure. Denote by \(\Gamma^{\varepsilon}\) the interface dividing \(\Omega_1^{\varepsilon}\), which is connected, and \(\Omega_2^{\varepsilon}\), which is disconnected. Denote by \(n_i\), the unitary outward normal to \(\Omega_i^{\varepsilon}\). For a given \(f \in L^2(\Omega)\) the authors study the limit behaviour, as \(\varepsilon \to 0\), of the following problems \[ \begin{alignedat}{2}2 -\text{div} (A^{\varepsilon} \nabla u^{\varepsilon}) &= f^{\varepsilon} && \quad\text{ in } \, \Omega_1^{\varepsilon} \cup \Omega_2^{\varepsilon} ,\\ A_1^{\varepsilon} \nabla u_1^{\varepsilon} \cdot n_1& = - A_2^{\varepsilon} \nabla u_2^{\varepsilon} \cdot n_2 && \quad\text{ in } \Gamma^{\varepsilon},\\ A_1^{\varepsilon} \nabla u_1^{\varepsilon} \cdot n_1 &= - \varepsilon^{\gamma} h^{\varepsilon} (u_1^{\varepsilon} - u_2^{\varepsilon}) && \quad\text{ in } \Gamma^{\varepsilon} ,\\ u^{\varepsilon}&= 0 && \quad\text{ in } \partial \Omega, \end{alignedat} \] where \(A_i^{\varepsilon}\) and \(u_i^{\varepsilon}\) denote the restriction to \(\Omega_i^{\varepsilon}\), of respectively \(A^{\varepsilon}\) and \(u^{\varepsilon}\). The matrix \(A^{\varepsilon}\) is defined as \(A^{\varepsilon}(x) = A({\varepsilon}^{-1}x)\) with \(A\) positive definite and with \(L^{\infty}\) coefficients and \(h^{\varepsilon}(x) = h ({\varepsilon}^{-1}x)\) with \(h^{\varepsilon} \in L^{\infty}(\Gamma^{\varepsilon})\) and \(h(y) > c > 0\). The parameter \(\gamma\) is chosen greater than \(-1\), but the situations \(-1 < \gamma \leqslant 1\) and \(\gamma > 1\) are treated separately: for \(-1 < \gamma \leqslant 1\) \(f^{\varepsilon}=f\) for every \(\varepsilon > 0\), for \(\gamma > 1\) the function \(f\) is renormalized in a suitable way in \(\Omega_2^{\varepsilon}\) (to avoid unboundedness of the solutions) and \(f^{\varepsilon} \to \theta f\) weakly in \(L^2(\Omega)\), where \(\theta\) is the weak limit of \(\chi_{\Omega_1^{\varepsilon}}\).

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J20 Variational methods for second-order elliptic equations
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References:

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