JournalsrlmVol. 33, No. 1pp. 39–64

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

We investigate a degenerating elliptic problem in a multi-structure Ωε\Omega_\varepsilon of R3\mathbb{R}^3, in the framework of the thermal stationary conduction with highly contrasting diffusivity. Precisely, Ωε\Omega_\varepsilon consists of a fixed basis Ω\Omega^- surmounted by a thin cylinder Ωε+\Omega_\varepsilon^+ with height 11 and cross-section with a small diameter of order ε\varepsilon. Moreover, Ωε+\Omega^+_\varepsilon contains a cylindrical core, always with height 11 and cross-section with diameter of order ε\varepsilon, with conductivity of order 11, surrounded by a ring with conductivity of order ε2\varepsilon^2. Also Ω\Omega^- has conductivity of order ε2\varepsilon^2. By assuming that the temperature is zero on the top and on the bottom of the boundary of Ωε\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 ε\varepsilon vanishes, boils down to two uncoupled problems: one in Ω\Omega^- and one in Ω1+\Omega^+_1, and the problem in Ω1+\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