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

Kirill Cherednichenko; Serena D'Onofrio

doi:10.1007/s10958-018-3887-9

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

Kirill Cherednichenko, Serena D'Onofrio

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Abstract

For an arbitrary periodic Borel measure μ we prove order O(ε) operator-norm resolvent estimates for the solutions to scalar elliptic problems in L ²(ℝ ^d, dμ ^ε) with ε-periodic coefficients, ε > 0. Here, μ ^ε is the measure obtained by ε-scaling of μ. Our analysis includes the case of a measure absolutely continuous with respect to the standard Lebesgue measure, as well as the case of “singular” periodic structures (or “multistructures”), when μ is supported by lower-dimensional manifolds.

Original language	English
Pages (from-to)	558-572
Number of pages	15
Journal	Journal of Mathematical Sciences N.Y.
Volume	232
Issue number	4
Early online date	18 Jun 2018
DOIs	https://doi.org/10.1007/s10958-018-3887-9
Publication status	Published - 1 Jul 2018

ASJC Scopus subject areas

Statistics and Probability
General Mathematics
Applied Mathematics

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10.1007/s10958-018-3887-9

V_V_Zhikov_Issue_Cherednichenko_DOnofrio_submitted_v3
This is a post-peer-review, pre-copyedit version of an article published in Journal of Mathematical Sciences. The final authenticated version is available online at: https://doi.org/10.1007/s10958-018-3887-9
Accepted author manuscript, 354 KB

Cite this

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abstract = "For an arbitrary periodic Borel measure μ we prove order O(ε) operator-norm resolvent estimates for the solutions to scalar elliptic problems in L 2(ℝ d, dμ ε) with ε-periodic coefficients, ε > 0. Here, μ ε is the measure obtained by ε-scaling of μ. Our analysis includes the case of a measure absolutely continuous with respect to the standard Lebesgue measure, as well as the case of “singular” periodic structures (or “multistructures”), when μ is supported by lower-dimensional manifolds. ",

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N2 - For an arbitrary periodic Borel measure μ we prove order O(ε) operator-norm resolvent estimates for the solutions to scalar elliptic problems in L 2(ℝ d, dμ ε) with ε-periodic coefficients, ε > 0. Here, μ ε is the measure obtained by ε-scaling of μ. Our analysis includes the case of a measure absolutely continuous with respect to the standard Lebesgue measure, as well as the case of “singular” periodic structures (or “multistructures”), when μ is supported by lower-dimensional manifolds.

AB - For an arbitrary periodic Borel measure μ we prove order O(ε) operator-norm resolvent estimates for the solutions to scalar elliptic problems in L 2(ℝ d, dμ ε) with ε-periodic coefficients, ε > 0. Here, μ ε is the measure obtained by ε-scaling of μ. Our analysis includes the case of a measure absolutely continuous with respect to the standard Lebesgue measure, as well as the case of “singular” periodic structures (or “multistructures”), when μ is supported by lower-dimensional manifolds.

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

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Mathematical Foundations of Metamaterials: Homogenisation, Dissipation and Operator Theory

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

Abstract

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Mathematical Foundations of Metamaterials: Homogenisation, Dissipation and Operator Theory

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