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

In this paper, we consider a domain Ωε⊂RN, N≥2, with a very rough boundary depending on ε. For instance, if N=3Ωε has the form of a brush with an ε-periodic distribution of thin cylindrical teeth with fixed height and a small diameter of order ε. In Ωε we consider a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on ε, on the lateral boundary of the teeth. 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.