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 Γ with a periodic perforation of Rd, which looks like a sieve, given by Tε=⋃k∈Zd{εk+aεT} where T is a given compact set and aε≪ε is the size of the perforation in the ε-cell (0,ε)d⊂Rd. When ε tends to zero we establish uniform estimates for p-capacity, 1<p<d, of the set Γ∩Tε. Additionally, we prove that the intersections Γ∩{εk+aεT}k are uniformly distributed over Γ 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 Γ and the perforations, in a given bounded domain, is homogenizable when p<1+d4. This result is new even for the classical Laplace operator.