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Homogenization of a monotone problem in a domain with oscillating boundary. (English) Zbl 0942.35071
It is considered the following Neumann problem \[ -\text{div} (a(Du_h))+|u_h|^{p-2}u_h=f \quad \text{in }\Omega _h, \tag{1} \] \[ a(Du_h)\nu=0 \quad \text{on } \partial\Omega _h, \tag{2} \] in a domain \(\Omega _h\subset\Omega\subset \mathbb{R}^n\) \((n\geq 2)\), whose boundary \(\partial\Omega _h\) contains an oscillating part with respect to \(h\) when \(h\) tends to \(\infty\). The oscillating boundary is defined by a set of cylinders with axis \(0x_n\) that are \(h^{-1}-\)periodically distributed. In the problem (1), (2) \(p\) is a given number in \(]1,+\infty[, f\) a given function in \(L^{\frac{p}{p-1}}(\Omega), a=(a_1,\ldots ,a_n)\) a monotone continuous function from \(\mathbb{R}^n\) to \(\mathbb{R}^n\) satisfying the following conditions: \[ \exists\alpha\in ]0,+\infty[: \qquad \alpha|\xi|^p\leq a(\xi)\xi \quad \forall\xi\in \mathbb{R}^n, \] \[ \exists\beta ,\gamma\in ]0,+\infty[: \qquad |a(\xi)|\leq\beta +\gamma |\xi|^{p-1}\quad\forall\xi\in \mathbb{R}^n, \] and \(\nu\) denotes the exterior unit normal to \(\Omega _h\). The asymptotic behaviour of the solution \(u_h\) of the problem under consideration, as \(h\) diverges, is studied. It is proved that the limit problem in the domain corresponding to the oscillating boundary identifies with a diffusion operator with respect to \(x_n\) coupled with an algebraic problem for the limit fluxes.

35J60 Nonlinear elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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