On Homogenization of Nonlinear Robin Type Boundary Conditions for the n-Laplacian in n-Dimensional Perforated Domains

We address the homogenization of a boundary value problem posed in perforated media for the p-Laplacian. We consider p = n, that is the n-Laplace operator in a perforated domain of ℝ n $$\mathbb{R}^{n}$$ , n ≥ 3, while the flux (associated with the n-Laplacian) on the boundary of the perforations is given by a negative, nonlinear monotonic function of the solution which is multiplied by a parameter which can be very large compared with the periodicity of the structure O(ɛ). A certain non-periodical distribution of the perforations is allowed, while the assumption that their size is much smaller than the periodicity scale ɛ is performed. We consider different relations between the parameters of the problem and, as ɛ → 0, we obtain all the possible homogenized problems. For certain of these relations, in the average constants of the problem, the perimeter of the perforations appears for any shape.