Homogenization of diffusion problems with a nonlinear interfacial resistance
The authors study the asymptotic behaviour as \(\varepsilon \to 0\) of the following problems given in an open bounded and with Lipschitz boundary set \(\Omega \subset {\mathbb R}^n\): \[ \begin{cases} - \operatorname{div} ( A^{\varepsilon} (x) \nabla u_1^{\varepsilon} (x) ) + h_1^{\varepsilon}(x, u_1^{\varepsilon}(x)) = f (x) & \text{in } \Omega_1^{\varepsilon} \\ - \operatorname{div} ( A^{\varepsilon} (x) \nabla u_2^{\varepsilon} (x) ) + h_2^{\varepsilon}(x, u_2^{\varepsilon}(x)) = f (x) & \text{in } \Omega_2^{\varepsilon} \\ A^{\varepsilon} (x) \nabla u_1^{\varepsilon} (x) \cdot n_1^{\varepsilon} = - A^{\varepsilon} (x) \nabla u_2^{\varepsilon} (x) \cdot n_2^{\varepsilon} & \text{in } \Gamma^{\varepsilon} \\ - A^{\varepsilon} (x) \nabla u_1^{\varepsilon} (x) \cdot n_1^{\varepsilon} = \varepsilon^{\gamma + 1} (x) h^{\varepsilon} (x, u_1^{\varepsilon} (x) - u_2^{\varepsilon}(x)) & \text{in } \Gamma^{\varepsilon} \\ u_1^{\varepsilon} = 0 & \text{in } \partial \Omega \cap \partial \Omega_1^{\varepsilon} \\ u_2^{\varepsilon} = 0 & \text{in } \partial \Omega \cap \partial \Omega_2^{\varepsilon} \end{cases} \] where \(\Omega_2^{\varepsilon}\) represents a structure of periodic and unvaried holes with Lipschitz boundary and \(\Omega_1^{\varepsilon} = \Omega \setminus \Omega_2^{\varepsilon}\), while \(\Gamma^{\varepsilon}\) represents the boundary of \(\Omega_2^{\varepsilon}\). \(A^{\varepsilon}\) is a matrix defined as \[ A^{\varepsilon} (x) := A \big( x / \varepsilon) \] where \(A\) is a matrix whose entries are functions in \(L^{\infty} (\Omega)\) and satisfies \[ \big( A (x) \lambda, \lambda \big) \geqslant \alpha |\lambda|^2 \, , \qquad \big| A (x) \lambda \big| \leqslant \beta |\lambda| \] for every \(\lambda \in {\mathbb R}^n\) and for two given positive constant \(\alpha\) and \(\beta\). The peculiarity and the difficulty rely on the fact that \(h\), \(h_1\), \(h_2\) may be non-linear functions satisfying suitable assumptions.
scientific article; zbMATH DE number 6494628
scientific article