Limit models for thin heterogeneous structures with high contrast

Abstract 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 C ˆ n with cross-section of radius r n h n and its complementary annulus I ˆ n and we treat two complementary cases. In the first case we consider a low conductivity of order δ n 2 in the core C ˆ n and a conductivity of order 1 in the annulus I ˆ n ; 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.