An uncoupled limit model for a high-contrast problem in a thin multi-structure
Umberto De Maio, Antonio Gaudiello, Ali Sili · Rendiconti Lincei Matematica e Applicazioni · 2022
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.