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

Abstract

We consider the intersection of a convex surface $\Ga$ with a periodic perforation of $\R^d$, which looks like a sieve, given by $T_\e = \bigcup_{k\in \Z^d}\{\e k+a_\e T\}$ where $T$ is a given compact set and $a_\e\ll \e$ is the size of the perforation in the $\e$-cell $(0, \e)^d\subset \R^d$. When $\e$ tends to zero we establish uniform estimates for $p$-capacity $1<p<d$ and discrepancy of distributions of intersection $\Ga\cap T_\e$. As an application one gets that the thin obstacle problem with the obstacle defined on the intersection of $\Ga$ and perforations, in given bounded domain, is homogenizable when $p<1+\frac d4$. This result is new even for the classical Laplace operator.