Unfolding Operator Method for Thin Domains with a Locally Periodic Highly Oscillatory Boundary

Abstract

: We analyze the behavior of solutions of the Poisson equation with homogeneous Neumann boundary conditions in a two-dimensional thin domain which presents locally periodic oscillations at the boundary. The oscillations are such that both the amplitude and period of the oscillations may vary in space. We obtain the homogenized limit problem and a corrector result by extending the unfolding operator method to the case of locally periodic media. We emphasize the fact that the techniques developed in this paper can be adapted to other locally periodic cases like reticulated or perforated domains where the period may be space-dependent.

Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

“…This is a preliminary step towards a complete generalization of the papers of Bayada and Chambat [6,7] in order to consider rough surfaces of type η ε h(x , x /ε) (locally periodic oscillatory boundaries), which are more practical from the engineering point of view. We think that this could be successfully managed by an adaptation of the recent version of the unfolding method introduced by Arrieta and Villanueva-Pesqueira [4], which will be object of a future study.…”

O p-Laplaciano em domínios finos oscilantes

“…We suppose g : R → R strictly positive and L-periodic satisfying (3), but now, just being lower semicontinuous. Here, the asymptotic analysis is performed using the monotone operator techniques and the unfolding method approach [10,11,24,25,26] which is a more general and modern framework than that one used in Chapter 1.…”

“…and such that G, ∂ x G and ∂ y G are uniformly bounded in (ξ i −1 , ξ i ) × R and have limits when we approach ξ i −1 and ξ i . As before, we also suppose there exist two constants G 0 and G 1 such that In this case, besides monotone operator techniques, we use the periodic and the locally periodic unfolding methods respectively from [10,11]. It is worth to point out here that we also need to generalize a perturbation result from [6] which allows us to pass to the limit in the equation.…”

“…From pioneering works to recent ones we mention [37,35,59,44,31,45,11,10] concerned with elliptic and parabolic equations, as well as [12,36,14,30,38,43,16,2] where the authors considered Stokes and Navier-Stokes equations from fluid mechanics. The second author also have studied different classes of thin domains problems for elliptic and parabolic equations.…”

“…We will obtain different limit problems according to this three cases. Here we will combine techniques such as unfolding operator methods for thin domains, which were developed in [11,10], as well as, those ones presented in [27,29] in order to analyze monotone operators in perforated domains. Furthermore, we will also obtain corrector results for each case considered here.…”