Limit models for thin heterogeneous structures with high contrast

A. Gaudiello (1) , Ali Sili (2)

1 Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
2 I2M - Institut de Mathématiques de Marseille

We investigate two linear conductivity problems, with strongly contrasting conductivity, in a thin heterogeneous cylinder with a small cross-section of radius h n , n in N. In this cylinder we distinguish an inner cylindrical core Ĉn with cross-section of radius r n << h n and its complementary annulus În and we treat two complementary cases. In the first case we consider a low conductivity of order δ 2 n in the core Ĉn and a conductivity of order 1 in the annulus În ; the opposite situation in the second case. We study the asymptotic behavior of these problems with three small parameters: h n , r n , and δ n , as h n → 0, r n → 0, r n h n → 0, and δ n → 0. In the first case we prove that the inner core has not any influence on the limit behavior. In the second case, we pinpoint three different limit regimes depending on the ratio μ = lim n δ n h n , according to μ = 0, 0 < μ < +∞, or μ = +∞. We obtain L 2-strong convergence for the solution and its gradient. We examine the limit problems and compare them with other models.

Mots clés: Correctors; Contrasting conductivity; Thin structure; 35J70

Domaines: Equations aux dérivées partielles [math.AP]

A. Gaudiello, Ali Sili. Limit models for thin heterogeneous structures with high contrast. Journal of Differential Equations, 2021, 302, pp.37-63. ⟨10.1016/j.jde.2021.08.032⟩. ⟨hal-03662171⟩