Operator-norm convergence estimates for elliptic homogenisation problems on periodic singular structures
Kirill Cherednichenko, Serena D’Onofrio · arXiv (Cornell University) · 2018
For a an arbitrary periodic Borel measure $\\mu$, we prove order $O(\\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\\mathbb R}^d, d\\mu^\\varepsilon)$ with $\\varepsilon$-periodic coefficients, $\\varepsilon>0.$ Here $\\mu^\\varepsilon$ is the measure obtained by $\\varepsilon$-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.