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

Abstract

We consider a domain $\Omega_\varepsilon\subset\mathbb{R}^N$, $N\geq2$, with a very rough boundary depending on~$\varepsilon$. For instance, if $N=3$ the domain $\Omega_\varepsilon$ has the form of a brush with an $\varepsilon$-periodic distribution of thin cylinders with fixed height and a small diameter of order $\varepsilon$. In $\Omega_\varepsilon$ a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on $\varepsilon$, on the lateral boundary of the cylinders is considered. We study the asymptotic behavior of this problem, as $\varepsilon$ 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.