We investigate a degenerating elliptic problem in a multi-structure of , in the framework of the thermal stationary conduction with highly contrasting diffusivity. Precisely, consists of a fixed basis surmounted by a thin cylinder with height and cross-section with a small diameter of order . Moreover, contains a cylindrical core, always with height and cross-section with diameter of order , with conductivity of order , surrounded by a ring with conductivity of order . Also has conductivity of order . By assuming that the temperature is zero on the top and on the bottom of the boundary of , 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 vanishes, boils down to two uncoupled problems: one in and one in , and the problem in is nonlocal. Moreover, a corrector result is obtained.
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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. 39–64DOI 10.4171/RLM/963