Homogenization for the \(p\)-Laplace operator and nonlinear Robin boundary conditions in perforated media along \((n - 1)\)-dimensional manifolds
From the authors' introduction: We study a homogenization problem for the \(p\)-Laplace operator posed in a domain \(\Omega_\varepsilon\) which is obtained by removing balls \(G_\varepsilon\), the cavities, of diameter \(O(\varepsilon^\alpha)\) from a fixed domain \(\Omega\) of \(\mathbb{R}^n\). The balls are \(\varepsilon\)-periodically distributed along a \((n-1)\)-dimensional manifold and nonlinear Robin type conditions are imposed on the boundary of the cavities with a large adsorption constant \(\varepsilon^{-k}\). Here \(\varepsilon\) denotes a small parameter and we shall let \(\varepsilon\to 0\), \(\alpha> 1\) and \(k>0\). For the critical case \(\alpha={n-1\over n-p}\) and \(k={(p-1)(n-1)\over n-p}\) with \(p\) and \(n\) fixed, \(p\in[2, n)\), the homogenized problem is obtained. This problem has a nonlinear transmission condition which depends on the nonlinearity arising both in the equation and on the boundary conditions. It should be noted that the results in this paper extend those by \textit{M. Lobo} et al. [ibid. 83, No. 1, 34--38 (2011); translation from Dokl. Akad. Nauk 436, No. 2, 163--167 (2011; Zbl 1233.35075)] adding extra nonlinearity from the operator.
scientific article; zbMATH DE number 6336259