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