The $$p\,$$-Laplacian equation in a rough thin domain with terms concentrating on the boundary

Abstract

In this work we use reiterated homogenization and unfolding operator approach to study the asymptotic behavior of the solutions of the p-Laplacian equation with Neumann boundary conditions set in a rough thin domain with concentrated terms on the boundary. We study weak, resonant and high roughness, respectively. In the three cases, we deduce the effective equation capturing the dependence on the geometry of the thin channel and the neighborhood where the concentrations take place.

The p-Laplacian in thin channels with locally periodic roughness and different scales*

“…where g is a positive, bounded and periodic function satisfying some regularity hypothesis and ε > 0 is a small parameter which goes to zero. Thereby, in the limit ε → 0, the open set Q ε degenerates to the unit interval presenting oscillatory behaviour on the upper boundary (see for instance [1,3,5,[18][19][20][21][22] where similar approach are performed). The periodic rough boundary considered above is certainly a first step, but usually not enough, since most of the irregularities present in real applications are not periodic.…”