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 Ω e ⊂ R N , N ≥ 2 , with a very rough boundary depending on e. For instance, if N = 3 Ω e has the form of a brush with an e-periodic distribution of thin cylindrical teeth with fixed height and a small diameter of order e. In Ω e we consider a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on e, on the lateral boundary of the teeth. We study the asymptotic behavior of this problem, as e 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.