# zbMATH — the first resource for mathematics

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.

##### MSC:
 35J60 Nonlinear elliptic equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
Full Text: