The p-Laplacian in thin channels with locally periodic roughness and different scales* *Nakasato partially supported by CNPq 141675/2015-2 (Brazil), INCT/CAPES 88887.507830/2020-00 (Brazil) and Croatian Science Foundation under the project AsAn (IP-2018-01-2735). Pereira partially supported by CNPq 308950/2020-8, FAPESP 2020/04813-0 and 2020/14075-6 (Brazil), PID2019-103860GB-I00 MICINN (Spain) and Croatian Science Foundation under the project MultiFM IP-2019-04-1140 (Croatia).

Abstract

In this work we analyse the asymptotic behaviour of the solutions of the p-Laplacian equation with homogeneous Neumann boundary conditions posed in bounded thin domains as and for some α > 0. We take a smooth function , L-periodic in the second variable, which allows us to consider locally periodic oscillations at the upper boundary. The thin domain situation is established passing to the limit in the solutions as the positive parameter ɛ goes to zero and we determine the limit regime for three case: α < 1, α = 1 and α > 1.