Homogenization of a mixed boundary-value problem in a domain with anisotropic fractal perforation

We carry out a homogenization of a mixed boundary-value problem for a scalar elliptic equation in a rectangle with anisotropic fractal perforation, namely, the (small) size of holes is preserved in one direction, whereas it is reduced in the other when moving away from the base of the rectangle. Neumann conditions are imposed on the boundaries of the holes. A specific feature of the asymptotic constructions is the presence of several boundary layers. Explicit formulae are obtained for the homogenized differential operator and asymptotically exact error estimates are derived, and the smallness of the majorant is related to the smoothness property of the right-hand side with respect to the slow variable in the scale of Sobolev–Slobodetskii spaces.