Homogenization of the parabolic equation with periodic coefficients at the edge of a spectral gap

Akhmatova, A. R.; Aksenova, E. S.; Sloushch, V. A.; Suslina, T. A.

doi:10.1080/17476933.2021.1947259

Cited by 5 publications

(3 citation statements)

References 14 publications

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“…For this, let us introduce the objects associated with the spectral resolution of operator (2.7). We follow the papers [28][29][30], see also [4, Chapter 2, § § 1,2]. In L 2 (0, 1), consider the family of quadratic forms…”

Section: Spectral Decomposition Of Operator (27)mentioning

confidence: 99%

“…Now, let us discuss error estimates for high-frequency homogenization. This topic has been studied in [28][29][30] in the one-dimensional case (d = 1) and in [31,32] in the case of arbitrary dimension d. It is well-known that the spectrum of A has a band structure and may have gaps. For the sake of simplicity, we consider the case where d = 1 and Γ = Z; in this case we shall use the notation A ε for operator (1.12).…”

Section: Introductionmentioning

confidence: 99%

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High-frequency homogenization of nonstationary periodic equations

Dorodnyi¹

2022

Preprint

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We consider an elliptic differential operatorFor the nonstationary Schrödinger equation with the Hamiltonian A ε and for the hyperbolic equation with the operator A ε , analogs of homogenization problems, related to the edges of the spectral bands of the operator A ε , are studied (the so called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L 2 (R)-norm for small ε are obtained.

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Section: Spectral Decomposition Of Operator (27)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-frequency homogenization of nonstationary periodic equations

Dorodnyi¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Now, let us discuss error estimates for high-frequency homogenization. This topic has been studied in [32][33][34][35][36] in the one-dimensional case (d = 1) and in [37][38][39] in the case of arbitrary dimension d. It is well-known that the spectrum of A has a band structure and may have gaps. For the sake of simplicity, we consider the case where d = 1 and Γ = Z; in this case we shall use the notation A ε for operator (0.12).…”

mentioning

confidence: 99%

High-energy homogenization of a multidimensional nonstationary Schrödinger equation

Dorodnyi¹

2023

Preprint

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In L 2 (R d ), we consider an elliptic differential operator A ε = − div g(x/ε)∇ + ε −2 V (x/ε), ε > 0, with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian A ε , analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator A 1 are studied (the so called high-energy homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L 2 (R d )-norm for small ε are obtained.

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High-frequency homogenization of nonstationary periodic equations

Dorodnyi

2023

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Homogenization of the parabolic equation with periodic coefficients at the edge of a spectral gap

Cited by 5 publications

References 14 publications

High-frequency homogenization of nonstationary periodic equations

High-frequency homogenization of nonstationary periodic equations

High-energy homogenization of a multidimensional nonstationary Schrödinger equation

High-frequency homogenization of nonstationary periodic equations

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