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

For a an arbitrary periodic Borel measure $μ$, we prove order $O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\mathbb R}^d, dμ^\varepsilon)$ with $\varepsilon$-periodic coefficients, $\varepsilon>0.$ Here $μ^\varepsilon$ is the measure obtained by $\varepsilon$-scaling of $μ.$ 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 $μ$ is supported by lower-dimensional manifolds.

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