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Seminário Luiz Adauto Medeiros de Análise/EDP - 27/01/2023

23 01 Seminário Luiz Adauto Medeiros NoticiaTítulo: Boundary homogenization problems with high contrasts: the elasticity system & the local problems

Data: 27/01/2023
Horário: 11:00h
Local: C-116
Palestrante: María Eugenia Pérez Martínez (Universidad de Cantabria)

Resumo: We consider the homogenization problem for the elasticity operator posed in a bounded domain of the upper half-space, a part of its boundary being in contact with the plane. We assume that this surface is free outside small regions in which we impose Robin-Winkler boundary conditions linking stresses and displacements by means of a symmetric and positive definite matrix and a reaction parameter. These small regions are periodically placed along the plane while its size is much smaller than the period. We look at the asymptotic behaviour of spectrum and provide all the possible spectral homogenized problems depending on certain asymptotic relations between the period, the size of the regions and the reaction-parameter. We state the convergence of the eigenelements, as the period tends to zero, which deeply involves the corresponding microscopic stationary problems obtained by means of asymptotic expansions.
We compare results and techniques with those for the Laplace operator and outline some possible extensions (under consideration) of the problem.

Some references:

[1] D. Gómez, S.A. Nazarov, ; M.-E. Pérez-Martínez. Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. Journal of Elasticity, 2020, V. 142, p. 89-120.

[2] D. Gómez, S.A. Nazarov ; M.-E. Pérez-Martínez. Spectral homogenization problems in linear elasticity with large reaction terms concentrated in small regions of the boundary. In: Computational and Analytic Methods in Science and Engineering. Birkäuser, Springer, N.Y., 2020, pp. 121-143

[3] D. Gómez; M.-E. Pérez-Martínez. Boundary homogenization with large reaction terms on a strainer-type wall. Z. Angew. Math. Phys. Vol. 73, 28p 2022.

[4] M.-E. Pérez-Martínez. Homogenization for alternating boundary conditions with large reaction terms concentrated in small regions. In: Emerging problems in the homogenization of Partial Differential Equations. ICIAM2019 SEMA SIMAI Springer Series 10, 2021, pp. 37-57.

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