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

### Umberto De Maio

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

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

Aix-Marseille Université, France

## Abstract

We investigate a degenerating elliptic problem in a multi-structure $Ω_{ε}$ of $R_{3}$, in the framework of the thermal stationary conduction with highly contrasting diffusivity. Precisely, $Ω_{ε}$ consists of a fixed basis $Ω_{−}$ surmounted by a thin cylinder $Ω_{ε}$ with height $1$ and cross-section with a small diameter of order $ε$. Moreover, $Ω_{ε}$ contains a cylindrical core, always with height $1$ and cross-section with diameter of order $ε$, with conductivity of order $1$, surrounded by a ring with conductivity of order $ε_{2}$. Also $Ω_{−}$ has conductivity of order $ε_{2}$. By assuming that the temperature is zero on the top and on the bottom of the boundary of $Ω_{ε}$, 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 $ε$ vanishes, boils down to two uncoupled problems: one in $Ω_{−}$ and one in $Ω_{1}$, and the problem in $Ω_{1}$ 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