Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization
Aram Karakhanyan, Martin Strömqvist · Calculus of Variations and Partial Differential Equations · 2016
We consider the intersection of a convex surface $$\Gamma $$ with a periodic perforation of $$\mathbb {R}^d$$ , which looks like a sieve, given by $$T_\varepsilon = \bigcup _{k\in \mathbb {Z}^d}\{\varepsilon k+a_\varepsilon T\}$$ where T is a given compact set and $$a_\varepsilon \ll \varepsilon $$ is the size of the perforation in the $$\varepsilon $$ -cell $$(0, \varepsilon )^d\subset \mathbb {R}^d$$ . When $$\varepsilon $$ tends to zero we establish uniform estimates for p-capacity, $$1<p<d$$ , of the set $$\Gamma \cap T_\varepsilon $$ . Additionally, we prove that the intersections $$\Gamma \cap \{\varepsilon k+a_\varepsilon T\}_k$$ are uniformly distributed over $$\Gamma $$ 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 $$\Gamma $$ and the perforations, in a given bounded domain, is homogenizable when $$p<1+\frac{d}{4}$$ . This result is new even for the classical Laplace operator.