Homogenization of a nonlinear monotone problem with nonlinear Signorini boundary conditions in a domain with highly rough boundary

Abstract

We consider a domain Ωε⊂Rⁿ, N≥2, with a very rough boundary depending on ~ε. For instance, if N=3 the domain Ωε has the form of a brush with an ε-periodic distribution of thin cylinders with fixed height and a small diameter of order ε. In Ωε a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on ε, on the lateral boundary of the cylinders is considered. We study the asymptotic behavior of this problem, as ε vanishes, i.e. when the number of thin attached cylinders increases unboundedly, while their cross sections tend to zero. We identify the limit problem which is a nonstandard homogenized problem. Namely, in the region filled up by the thin cylinders the limit problem is given by a variational inequality coupled to an algebraic system.