Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account

Sloushch, V. A.; Suslina, T. A.

doi:10.1134/s0016266320030077

Cited by 7 publications

(3 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years further progress has been achieved in this topic. We quote here [26], where operator estimates were obtained for the Stokes system, [71,35,39,40,53,38,67,68] that dealt with elliptic operators of higher order in R d , and [62,63,66], where the higher order elliptic operators in a bounded domain were considered.…”

mentioning

confidence: 99%

On operator estimates in homogenization of non-local operators of convolution type

Piatnitski¹,

Sloushch²,

Suslina³

et al. 2022

Preprint

View full text Add to dashboard Cite

The paper studies a bounded symmetric operator A ε in L 2 (R d ) withhere ε is a small positive parameter. It is assumed that a(x) is a non-negative L 1 (R d ) function such that a(−x) = a(x) and the momentsIt is also assumed that µ(x, y) is Z d -periodic both in x and y function such that µ(x, y) = µ(y, x) and 0 < µ − µ(x, y) µ + < ∞. Our goal is to study the limit behaviour of the resolvent (A ε + I) −1 , as ε → 0. We show that, as ε → 0, the operator (A ε + I) −1 converges in the operator norm in L 2 (R d ) to the resolvent (A 0 +I) −1 of the effective operator A 0 being a second order elliptic differential operator with constant coefficients of the form A 0 = − div g 0 ∇. We then obtain sharp in order estimates of the rate of convergence. R d a(z) dz = 1. On the other hand, each point of γ has a random life time, and the intensity of death is m(x) > 0. In the general case the intensities of birth and death might depend on

show abstract

mentioning

confidence: 99%

On operator estimates in homogenization of non-local operators of convolution type

Piatnitski¹,

Sloushch²,

Suslina³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Another method to prove operator estimates in homogenization is based on the Floquet-Bloch transformation and spectral arguments. It is used for high order operators, e.g., in [2], [21], and [8]. The result recently announced in [8] is related to a fourth-order elliptic self-adjoint matrix differential operator A ε with ε-periodic coefficients which satisfies a certain factorization condition.…”

mentioning

confidence: 99%

“…It is used for high order operators, e.g., in [2], [21], and [8]. The result recently announced in [8] is related to a fourth-order elliptic self-adjoint matrix differential operator A ε with ε-periodic coefficients which satisfies a certain factorization condition. In [8], we find resolvent approximations of the same order of smallness in error as in Theorem 5.2, but they look rather differently compared to that of (2.18).…”

mentioning

confidence: 99%

Improved approximations of resolvents in homogenization of fourth-order operators with periodic coefficients

Pastukhova

2021

Preprint

View full text Add to dashboard Cite

In the whole space R d , d ≥ 2, we study homogenization of a divergence form elliptic fourth-order operator A ε with measurable ε-periodic coefficients, where ε is a small parameter. For the resolvent (A ε + 1) −1 , acting as an operator from L 2 to H 2 , we find an approximation with remainder term of order O(ε 2 ) as ε → 0. Relying on this result, we construct the resolvent approximation with remainder of order O(ε 3 ) in the operator L 2 -norm. We employ two-scale expansions that involve smoothing.

show abstract

Homogenization of the Higher-Order Parabolic Equations with Periodic Coefficients

Miloslova,

Suslina

2023

J Math Sci

View full text Add to dashboard Cite

Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account

Cited by 7 publications

References 2 publications

On operator estimates in homogenization of non-local operators of convolution type

On operator estimates in homogenization of non-local operators of convolution type

Improved approximations of resolvents in homogenization of fourth-order operators with periodic coefficients

Homogenization of the Higher-Order Parabolic Equations with Periodic Coefficients

Contact Info

Product

Resources

About