Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization

We consider the intersection of a convex surface [Formula: see text] with a periodic perforation of [Formula: see text] , which looks like a sieve, given by [Formula: see text] where T is a given compact set and [Formula: see text] is the size of the perforation in the [Formula: see text] -cell [Formula: see text] . When [Formula: see text] tends to zero we establish uniform estimates for p-capacity, [Formula: see text] , of the set [Formula: see text] . Additionally, we prove that the intersections [Formula: see text] are uniformly distributed over [Formula: see text] and give estimates for the discrepancy of the distribution. As an application we show that the thin obstacle problem with the obstacle defined on the intersection of [Formula: see text] and the perforations, in a given bounded domain, is homogenizable when [Formula: see text] . This result is new even for the classical Laplace operator.