Homogenization in perforated domains: a Stokes grill and an adsorption process

Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

“…For these relations, in general, there appears asymptotically a strange term in the transmission conditions on Σ for the derivatives of the solutions and u ϵ , respectively. The homogenization of the Stokes problem only depends on r ( ϵ ) and, as is well known in the literature (cf, eg, Gómez et al, Allaire, and Brillard), the strange term only appears for the critical size of the obstacles (that is, ), and it involves the so‐called Stokes capacity matrix, cf and .…”

“…For different relations between ϵ , β ( ϵ ), and r ( ϵ ), we obtain the homogenized models, cf . We refer to Gómez et al for a precise description of the physical model under consideration in the stationary case. See also Goncharenko and Khilkova, and references therein, in connection with these kinds of models in perforated domains over the whole volume.…”

“…“The most critical situation” occurs when simultaneously we have the classical critical size of the obstacles and the critical adsorption parameter. In this case, the strange term is defined through the “average” expression , which is referred to as (cf Gómez et al) extended capacity , and involves the nonlinear function σ and the parameter β 0 , cf , relating the ratio of the obstacles and the adsorption: see the homogenized problem and the bi‐parametric family of local problems . In the case of very big obstacles, the strange term contains the area of the obstacles in the averaged constants accompanying σ : cf the homogenized problem .…”

“…Let us mention that stationary problems associated with our model have been addressed in Gómez et al and Brillard et al, where an extensive bibliography on the model and previous works is provided. In this paper, we give results that extend and complement those in Brillard et al and Gómez et al to arbitrary shapes of the obstacles and to the dimension n > 3, respectively. Further specifying, Brillard et al address the convergence in the most critical situation of the time‐independent scalar problem for a domain perforated by balls.…”

A nonlinear problem for the Laplace equation with a degenerating Robin condition

“…An extension to nonlinear equation has been considered, for example, in Berestycki and Wei . Degenerating nonlinear Robin conditions in the frame of homogenization problems have been studied by Gómez et al Singularly perturbed boundary conditions for the Maxwell equations have been analyzed, for example, in Ammari and Nédélec . Moreover, Schmidt and Hiptmair have exploited integral equation methods for singularly perturbed boundary conditions in the frame of transmission problems.…”