Correcting Terms for Perforated Media by Thin Tubes with Nonlinear Flux and Large Adsorption Parameters

Abstract

We obtain corrector terms for homogenization problems in perforated media. The perforations are thin cylindrical tubes, periodically distributed over a fixed domain of the three dimensional space. The operator under consideration is the Laplacian and we impose nonlinear Robin type boundary conditions on the boundary of the cavities and Dirichlet condition on the rest of the boundary. The period of the structure is given by a small parameter that converges towards zero. The diameter of the transverse sections of the tubes is of an order of magnitude much smaller than the period. Also a very large parameter (compared with the period) arises in the Robin conditions: the adsorption constant. Depending on the different values/relations between the three parameters (periodicity, diameter and adsorption) different homogenized problems have been obtained in [D. Gomez, M. Lobo, E. Perez, T.A. Shaposhnikova, M.N. Zubova, On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems, Math. Meth. Appl. Sci., DOI:10.1002/mma.3246], where convergences for solutions hold in the weak topology of the corresponding Sobolev spaces. The results in this chapter improve these convergences providing estimates for convergence rates.