An uncoupled limit model for a high-contrast problem in a thin multi-structure (Q2154794)

Summary: We investigate a degenerating elliptic problem in a multi-structure \(\Omega_\varepsilon\) of \(\mathbb{R}^3\), in the framework of the thermal stationary conduction with highly contrasting diffusivity. Precisely, \( \Omega_\varepsilon\) consists of a fixed basis \(\Omega^-\) surmounted by a thin cylinder \(\Omega_\varepsilon^+\) with height \(1\) and cross-section with a small diameter of order \(\varepsilon \). Moreover, \( \Omega^+_\varepsilon\) contains a cylindrical core, always with height \(1\) and cross-section with diameter of order \(\varepsilon \), with conductivity of order \(1\), surrounded by a ring with conductivity of order \(\varepsilon^2\). Also \(\Omega^-\) has conductivity of order \(\varepsilon^2\). By assuming that the temperature is zero on the top and on the bottom of the boundary of \(\Omega_\varepsilon \), while the flux is zero on the remaining part of the boundary, under a suitable choice of the source term we prove that the limit problem, as \(\varepsilon\) vanishes, boils down to two uncoupled problems: one in \(\Omega^-\) and one in \(\Omega^+_1\), and the problem in \(\Omega^+_1\) is nonlocal. Moreover, a corrector result is obtained.