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

TL;DR

This paper provides uniform estimates for capacity and distribution discrepancy of convex surfaces intersecting sieve-like perforations, leading to homogenization results for thin obstacle problems in perforated domains.

Contribution

It introduces new uniform estimates for capacity and discrepancy in sieve-like perforations and applies these to establish homogenization of thin obstacle problems for the first time.

Findings

Established uniform estimates for p-capacity and discrepancy as perforation size tends to zero.

Proved homogenization of the thin obstacle problem under specific conditions on p.

Extended classical results to new settings involving convex surfaces and perforated domains.

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.