Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices
Using two-scale convergence techniques the authors consider the parabolic diffusion type equation \[ \begin{cases} \mu^\varepsilon \partial_t u_\varepsilon - \operatorname{div} \lambda^\varepsilon \operatorname{grad} u_\varepsilon = 0, & \text{ in }\Omega_{\mathrm{int}}^\varepsilon \cup \Omega_{\mathrm{out}}^\varepsilon \times (0,T);\\ \varepsilon \alpha \partial_t u_\varepsilon -\varepsilon \beta \Delta^B u_\varepsilon = [\lambda^\varepsilon \operatorname{grad} u_\varepsilon \cdot \nu_\varepsilon]& \text{ on }\Gamma^\varepsilon \times (0,T), \end{cases} \] subject to initial and Dirichlet boundary conditions on \(\partial\Omega\); \([\ ]\) denotes the jump across the interface \(\Gamma^\varepsilon\), and \(\Delta^B\) is the Laplace Beltrami operator on \(\Gamma^\varepsilon\). Here, \(\Omega=\Omega_{\mathrm{int}}^\varepsilon \cup \Omega_{\mathrm{out}}^\varepsilon \cup \Gamma^\varepsilon \subseteq \mathbb{R}^N\) is open, connected and bounded; \(\Gamma^\varepsilon\) is the boundary of the periodically distributed inclusions of size proportional to \(\varepsilon\) (gathered in the set \(\Omega_{\mathrm{int}}^\varepsilon \)) with outward unit normal \(\nu^\varepsilon\). Also, \(\Gamma^\varepsilon\) is \(\varepsilon\) far away from \(\partial\Omega\). The parameters \(\alpha\) and \(\beta\) are fixed, and the coefficient functions \(\lambda^\varepsilon\) and \(\mu^\varepsilon\) are constant on each \(\Omega_{\mathrm{int}}^\varepsilon \) and \(\Omega_{\mathrm{out}}^\varepsilon \). After showing a new two-scale convergence result particularly involving evaluations at an interface, the authors address the limit \(\varepsilon \to 0\) and compute the two-scale limit equation of the above equation. The authors solely focus on computing the two-scale convergence result and address the uniqueness and existence of the above equation elsewhere.
scientific article; zbMATH DE number 6932805
scientific article