Geodesic
Improved $L^2$-approximation of resolvents in homogenization of fourth order operators
Algebra i analiz, Tome 34 (2022) no. 4, pp. 74-106 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A 4th order elliptic operator $A_\varepsilon$ in the diverdence form acting in the entire space $\mathbb{R}^d$ and having $\varepsilon$-periodic coefficients is studied ($\varepsilon$ is a small parameter). An approximation for the resolvent $(A_\varepsilon+1)^{-1}$ is found with error estimate of order $\varepsilon^3$ in the operator $(L^2{\to}L^2)$-norm. The method of double-scale approximation with a generalised shift in the form of smoothing is used.
Keywords: homogenization, error estimates, approximation of the resolvent, elliptic operator of the 4th order. 
@article{AA_2022_34_4_a3,
     author = {S. E. Pastukhova},
     title = {Improved $L^2$-approximation of resolvents in homogenization of fourth order operators},
     journal = {Algebra i analiz},
     pages = {74--106},
     year = {2022},
     volume = {34},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2022_34_4_a3/}
}
TY  - JOUR
AU  - S. E. Pastukhova
TI  - Improved $L^2$-approximation of resolvents in homogenization of fourth order operators
JO  - Algebra i analiz
PY  - 2022
SP  - 74
EP  - 106
VL  - 34
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/AA_2022_34_4_a3/
LA  - ru
ID  - AA_2022_34_4_a3
ER  - 
%0 Journal Article
%A S. E. Pastukhova
%T Improved $L^2$-approximation of resolvents in homogenization of fourth order operators
%J Algebra i analiz
%D 2022
%P 74-106
%V 34
%N 4
%U http://geodesic.mathdoc.fr/item/AA_2022_34_4_a3/
%G ru
%F AA_2022_34_4_a3
S. E. Pastukhova. Improved $L^2$-approximation of resolvents in homogenization of fourth order operators. Algebra i analiz, Tome 34 (2022) no. 4, pp. 74-106. http://geodesic.mathdoc.fr/item/AA_2022_34_4_a3/

[1] Zhikov V. V., Kozlov C. M., Oleinik O. A., Ngoan Kha Ten, “Usrednenie i $G$-skhodimost ellipticheskikh differentsialnykh operatorov”, Uspekhi mat. nauk, 34:5 (1979), 65–133 | MR | Zbl

[2] Veniaminov N. A., “Usrednenie periodicheskikh differentsialnykh operatorov vysokogo poryadka”, Algebra i analiz, 22:5 (2010), 69–103 | MR

[3] Kukushkin A. A., Suslina T. A., “Usrednenie ellipticheskikh operatorov vysokogo poryadka s periodicheskimi koeffitsientami”, Algebra i analiz, 28:1 (2016), 103–163

[4] Pastukhova S. E., “Operatornye otsenki usredneniya dlya ellipticheskikh uravnenii chetvertogo poryadka”, Algebra i analiz, 28:2 (2016), 204–226

[5] Pastukhova S. E., “Estimates in homogenization of higher-order elliptic operators”, Appl. Anal., 95:7 (2016), 1449–1466 | DOI | MR | Zbl

[6] Pastukhova S. E., “$L^2$-approksimatsii rezolventy v usrednenii ellipticheskikh operatora chetvertogo poryadka”, Mat. sb., 212:1 (2021), 119–142 | MR | Zbl

[7] Pastukhova S. E., “Approximation of resolvents in homogenization of higher order elliptic operators”, J. Math. Sci. (N.Y.), 251:6 (2020), 902–925 | DOI | MR | Zbl

[8] Pastukhova S. E., “Improved approximations of the resolvent in homogenization of fourth order operators”, J. Math. Sci. (N.Y.), 255:4 (2021), 488–502 | DOI | MR | Zbl

[9] Pastukhova S. E., “Improved approximations of the resolvent in homogenization of higher order operators”, J. Math. Sci. (N.Y.), 259:3 (2021), 230–243 | DOI | MR | Zbl

[10] Slousch V. A., Suslina T. A., “Usrednenie ellipticheskogo operatora chetvertogo poryadka s periodicheskimi koeffitsientami pri uchete korrektorov”, Funkts. anal. i ego pril., 54:3 (2020), 94–99 | MR

[11] Slousch V. A., Suslina T. A., “Usrednenie ellipticheskogo operatora chetvertogo poryadka s periodicheskimi koeffitsientami”, Sb. materialov mezhdunar. konf. KROMSh-2020, Poliprint, Simferopol, 2020, 186–188

[12] S. E. Pastukhova, “Improved approximations of resolvents in homogenization of higher order operators. The selfadjoint case”, J. Math. Sci. (N.Y.), 262:3 (2022), 312–328 | DOI | MR | Zbl

[13] Bakhvalov N. C., Panasenko G. P., Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR

[14] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993

[15] Zhikov V. V., Pastukhova S. E., “Ob operatornykh otsenkakh v teorii usredneniya”, Uspekhi mat. nauk, 71:3 (2016), 3–98 | MR

[16] Zhikov V. V., Pastukhova S. E., “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Physics, 12:4 (2005), 515–524 | MR | Zbl

[17] Zhikov V. V., Pastukhova S. E., “Estimates of homogenization for a parabolic equation with periodic coefficients”, Russ. J. Math. Physics, 13:4 (2006), 224–237 | DOI | MR | Zbl

[18] Pastukhova S. E., “$L^2$-estimates for homogenization of elliptic operators”, J. Math. Sci. (N.Y.), 244:4 (2020), 671–685 | DOI | MR | Zbl

[19] Pastukhova S. E., “$L^2$-approksimatsii rezolventy ellipticheskogo operatora v perforirovannom prostranstve”, Sovrem. mat. fundam. napravl., 66, no. 2, 2020, 314–334 | MR

[20] Pastukhova S. E., “Homogenization estimates for singularly perturbed operators”, J. Math. Sci. (N.Y.), 251:5 (2020), 724–747 | DOI | MR | Zbl

[21] Niu W., Yuan Y., “Convergence rate in homogenization of elliptic systems with singular perturbations”, J. Math. Phys., 60:11 (2019), 111509 | DOI | MR | Zbl

[22] Zhikov V. V., “Ob operatornykh otsenkakh v teorii usredneniya”, Dokl. RAN, 403:3 (2005), 305–308 | MR | Zbl

[23] Birman M. Sh., Suslina T. A., “Periodicheskie differentsialnye operatory vtorogo poryadka. Porogovye svoistva usredneniya”, Algebra i analiz, 15:5 (2003), 1–108

[24] Slousch V. A., Suslina T. A., “Porogovye approksimatsii rezolventy polinomialnogo operatornogo puchka”, Algebra i analiz, 33:2 (2021), 233–274

[25] Senik N. N., “Ob usrednenii nesamosopryazhennykh periodicheskikh ellipticheskikh operatorov v beskonechnom tsilindre”, Funkts. anal. i ego pril., 50:1 (2016), 85–89 | MR | Zbl

[26] Senik N. N., “Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder”, SIAM J. Math. Anal., 49:2 (2017), 874–898 | DOI | MR | Zbl

[27] Senik N. N., “Ob usrednenii nesamosopryazhennykh lokalno periodicheskikh ellipticheskikh operatorov”, Funkts. anal. i ego pril., 51:2 (2017), 92–96 | MR | Zbl

[28] Senik N. N., “Homogenization for locally periodic elliptic operators”, J. Math. Anal. Appl., 505:2 (2022), 125581 | DOI | MR | Zbl