Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

A. V. Kiselev; Luis O. Silva; Kirill Cherednichenko

doi:10.1134/S0001434622030051

Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

A. V. Kiselev, Luis O. Silva, Kirill Cherednichenko

Universidad Nacional Autónoma de México

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Abstract

Abstract: Using a generalization of the classical notion of Weyl m-function and related formulas for the resolvents of boundary-value problems, we analyze the asymptotic behavior of solutions to a “transmission problem” for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit problem of “electrostatic” type. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates. The approach developed in the paper is of a general nature and can thus be successfully applied in the study of other problems of the same type.

Original language	English
Pages (from-to)	373-387
Number of pages	15
Journal	Mathematical Notes
Volume	111
Issue number	3-4
Early online date	26 Apr 2022
DOIs	https://doi.org/10.1134/S0001434622030051
Publication status	Published - 30 Apr 2022

Bibliographical note

Funding Information:
The work of the first author was supported by the Russian Science Foundation under grant 20-11-20032. The work of the second author was supported by PASPA-DGAPA-UNAM during his sabbatical leave; he thanks the University of Bath for their hospitality. The work of the third author was supported by EPSRC under grants EP/L018802/2 and EP/V013025/1. The work of the second and third authors was also supported in part by the grant of CONACyT CF-2019 No. 304005. The second and third authors are grateful for the financial support of the Royal Society Newton Fund under the grant “Homogenisation of degenerate equations and scattering for new materials”.

Publisher Copyright:
© 2022, Pleiades Publishing, Ltd.

Keywords

extensions of symmetric operators; generalized boundary triples; boundary value problems; spectrum; transmission problems

ASJC Scopus subject areas

General Mathematics

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10.1134/S0001434622030051

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Cite this

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title = "Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions",

abstract = "Abstract: Using a generalization of the classical notion of Weyl m-function and related formulas for the resolvents of boundary-value problems, we analyze the asymptotic behavior of solutions to a “transmission problem” for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit problem of “electrostatic” type. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates. The approach developed in the paper is of a general nature and can thus be successfully applied in the study of other problems of the same type.",

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author = "Kiselev, {A. V.} and Silva, {Luis O.} and Kirill Cherednichenko",

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N1 - Funding Information: The work of the first author was supported by the Russian Science Foundation under grant 20-11-20032. The work of the second author was supported by PASPA-DGAPA-UNAM during his sabbatical leave; he thanks the University of Bath for their hospitality. The work of the third author was supported by EPSRC under grants EP/L018802/2 and EP/V013025/1. The work of the second and third authors was also supported in part by the grant of CONACyT CF-2019 No. 304005. The second and third authors are grateful for the financial support of the Royal Society Newton Fund under the grant “Homogenisation of degenerate equations and scattering for new materials”. Publisher Copyright: © 2022, Pleiades Publishing, Ltd.

PY - 2022/4/30

Y1 - 2022/4/30

N2 - Abstract: Using a generalization of the classical notion of Weyl m-function and related formulas for the resolvents of boundary-value problems, we analyze the asymptotic behavior of solutions to a “transmission problem” for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit problem of “electrostatic” type. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates. The approach developed in the paper is of a general nature and can thus be successfully applied in the study of other problems of the same type.

AB - Abstract: Using a generalization of the classical notion of Weyl m-function and related formulas for the resolvents of boundary-value problems, we analyze the asymptotic behavior of solutions to a “transmission problem” for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit problem of “electrostatic” type. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates. The approach developed in the paper is of a general nature and can thus be successfully applied in the study of other problems of the same type.

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JF - Mathematical Notes

IS - 3-4

ER -

Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Fingerprint

Quantitative tools for upscaling the micro-geometry of resonant media

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Mathematical Foundations of Metamaterials: Homogenisation, Dissipation and Operator Theory

Cite this

Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Quantitative tools for upscaling the micro-geometry of resonant media

Newton Mobility Grant -: Homogenisation of Degenerate Equations and Scattering for New Materials

Mathematical Foundations of Metamaterials: Homogenisation, Dissipation and Operator Theory

Cite this