Concentrated reaction terms on the boundary of rough domains for a quasilinear equation

Abstract

In this work we analyze the solutions of a p-Laplacian equation with homogeneous Neumann boundary conditions set in a family of rough domains with a nonlinear term concentrated on the boundary. At the limit, we get a nonlinear boundary condition capturing the oscillatory geometry of the strip where the reactions take place.

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

“…Remark 5.1. We point out that the results in [21] are proved in the unit interval. Here, we just rewrite it to (ξ i−1 , ξ i ).…”

“…Before proving the main result of this section, let us recall an important result proved, for instance, in [21]. It is concerned to the purely periodic thin domain situation.…”

“…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.…”

“…Therefore, analysing the asymptotic behaviour of models set on thin structures understanding how the geometry and the roughness affect the problem is of considerable current interest in applied science. In these directions, let us mention [7,10,15,21] and references therein.…”