Operator-norm convergence estimates for elliptic homogenisation problems on periodic singular structures

Abstract

For a an arbitrary periodic Borel measure $\mu$, we prove order $O(\epsilon )$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\mathbb{R}}^d,d{\mu }^{\epsilon })$ with $\epsilon$-periodic coefficients, $\epsilon >0.$ Here ${\mu }^{\epsilon }$ is the measure obtained by $\epsilon$-scaling of $\mu .$ Our analysis includes both the case of a measure absolutely continuous with respect to the standard Lebesgue measure and the case of "singular" periodic structures (or "multistructures"), when $\mu$ is supported by lower-dimensional manifolds.

16 pages