Thresholds in high contrast boundary homogenization problems on sieve-like structures

Resumo: We analyse the asymptotic behaviour of solutions of boundary homogenization problems for the elasticity system on sieve-like structures with rapidly alternating boundary conditions of Winkler-Robin type containing reaction parameters in small regions. These boundary conditions can be nonlinear and the parameters can range from very large to very small conditioning the averaged behaviour of the solutions.  Depending on the relations between the parameters (period, reaction and sizes of the reaction regions), the homogenization of linear boundary conditions can imply the same asymptotic  behaviour. We provide a general framework for different operators and state the spectral convergence for associated spectral problems.   The talk is based on joint works  [1-3] and  some recent works in progress.
1.- D. Gómez,  S.A. Nazarov and M.-E. Pérez-Martínez:  Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. J. Elasticity. 142, 89-120, 2020.
2.- D. Gómez and M.-E Pérez-Martínez:  Boundary homogenization with large reaction terms on a strainer-type wall.  ZAMP,  73:234, p. 1-28, 2022.
3.- D. Gómez and M.-E Pérez-Martínez:  Averaged reaction for nonlinear boundary conditions on a grill-type Winkler foundation. MMA J., to appear, 2024.

 

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