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

### Umberto De Maio

Università degli Studi di Napoli “Federico II”; and Istituto Nazionale di Alta Matematica, Italy### Antonio Gaudiello

Università degli Studi della Campania “Luigi Vanvitelli”, Caserta; and Istituto Nazionale di Alta Matematica, Italy### Ali Sili

Aix-Marseille Université, France

## Abstract

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.

## Cite this article

Umberto De Maio, Antonio Gaudiello, Ali Sili, An uncoupled limit model for a high-contrast problem in a thin multi-structure. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 33 (2022), no. 1, pp. 39–64

DOI 10.4171/RLM/963