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

Abstract

We investigate a degenerating elliptic problem in a multi-structure ${\Omega }_{\epsilon }$ of ${\mathbb{R}}^3$, in the framework of the thermal stationary conduction with highly contrasting diffusivity. Precisely, ${\Omega }_{\epsilon }$ consists of a fixed basis ${\Omega }^{-}$ surmounted by a thin cylinder ${\Omega }_{\epsilon }^{+}$ with height $1$ and cross-section with a small diameter of order $\epsilon$. Moreover, ${\Omega }_{\epsilon }^{+}$ contains a cylindrical core, always with height $1$ and cross-section with diameter of order $\epsilon$, with conductivity of order $1$, surrounded by a ring with conductivity of order ${\epsilon }^2$. Also ${\Omega }^{-}$ has conductivity of order ${\epsilon }^2$. By assuming that the temperature is zero on the top and on the bottom of the boundary of ${\Omega }_{\epsilon }$, 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 $\epsilon$ 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.