The \(p\)-Laplacian equation in thin domains: the unfolding approach. (English) Zbl 1455.35008
Summary: In this work we apply the so called Unfolding Operator Method to analyze the asymptotic behavior of the solutions of the \(p\)-Laplacian equation with Neumann boundary condition in a bounded thin domain of the type \(R^\varepsilon = \left\{ ( x , y ) \in \mathbb{R}^2 : x \in ( 0 , 1 ) \text{ and } 0 < y < \varepsilon g ( x / \varepsilon^\alpha ) \right\}\) where \(g\) is a positive periodic function. We study the three cases \(0 < \alpha < 1, \alpha = 1\) and \(\alpha > 1\) representing respectively weak, resonant and high oscillations at the top boundary. In the three cases we deduce the homogenized limit and obtain correctors.
MSC:
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35B25 | Singular perturbations in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J25 | Boundary value problems for second-order elliptic equations |
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