Homogenization of a Boundary Value Problem for the n-Laplace Operator on a n-Dimensional Domain with Rapidly Alternating Boundary Condition Type: The Critical Case

Abstract

We study the asymptotic behavior of the solution of a boundary value problem for the p-Laplace operator with rapidly alternating nonlinear boundary conditions posed on ε-periodically arranged subsets on the boundary of a domain Ω ⊂ ℝⁿ. We assume that p = n, construct a homogenized problem, and prove the weak convergence as ε → 0 of the solution of the original problem to the solution of the homogenized problem in the so-called critical case, which is characterized by the fact that the homogenization changes the character of nonlinearity of the boundary condition.