Research output: Contribution to journal › Article › peer-review
Operator estimates for homogenization of higher-order elliptic operators with periodic coefficients. / Слоущ, Владимир Анатольевич; Суслина, Татьяна Александровна.
In: St. Petersburg Mathematical Journal, Vol. 35, No. 2, 21.06.2024, p. 327-375.Research output: Contribution to journal › Article › peer-review
}
TY - JOUR
T1 - Operator estimates for homogenization of higher-order elliptic operators with periodic coefficients
AU - Слоущ, Владимир Анатольевич
AU - Суслина, Татьяна Александровна
PY - 2024/6/21
Y1 - 2024/6/21
N2 - In L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) , a strongly elliptic differential operator A ε {\mathcal {A}}_\varepsilon of order 2 p 2p is studied. Its coefficients are periodic and depend on x / ε \mathbf {x}/\varepsilon . The resolvent ( A ε + I ) − 1 ({\mathcal {A}}_\varepsilon +I)^{-1} is approximated in the operator norm on L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) : ( A ε + I ) − 1 = ( A 0 + I ) − 1 + ∑ j = 1 2 p − 1 ε j K j , ε + O ( ε 2 p ) . \begin{equation*} ({\mathcal {A}}_\varepsilon +I)^{-1} = ({\mathcal {A}}^0+I)^{-1} + \sum _{j=1}^{2p-1} \varepsilon ^{j} {\mathcal {K}}_{j,\varepsilon } + O(\varepsilon ^{2p}). \end{equation*} Here A 0 {\mathcal {A}}^0 is an effective operator with constant coefficients, and K j , ε {\mathcal {K}}_{j,\varepsilon } , j = 1 , … , 2 p − 1 j=1,\dots ,2p-1 , are appropriate correctors.
AB - In L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) , a strongly elliptic differential operator A ε {\mathcal {A}}_\varepsilon of order 2 p 2p is studied. Its coefficients are periodic and depend on x / ε \mathbf {x}/\varepsilon . The resolvent ( A ε + I ) − 1 ({\mathcal {A}}_\varepsilon +I)^{-1} is approximated in the operator norm on L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) : ( A ε + I ) − 1 = ( A 0 + I ) − 1 + ∑ j = 1 2 p − 1 ε j K j , ε + O ( ε 2 p ) . \begin{equation*} ({\mathcal {A}}_\varepsilon +I)^{-1} = ({\mathcal {A}}^0+I)^{-1} + \sum _{j=1}^{2p-1} \varepsilon ^{j} {\mathcal {K}}_{j,\varepsilon } + O(\varepsilon ^{2p}). \end{equation*} Here A 0 {\mathcal {A}}^0 is an effective operator with constant coefficients, and K j , ε {\mathcal {K}}_{j,\varepsilon } , j = 1 , … , 2 p − 1 j=1,\dots ,2p-1 , are appropriate correctors.
UR - https://www.mendeley.com/catalogue/92b10fa8-ab9b-3d1c-b6c1-efe96125913c/
U2 - 10.1090/spmj/1807
DO - 10.1090/spmj/1807
M3 - Article
VL - 35
SP - 327
EP - 375
JO - St. Petersburg Mathematical Journal
JF - St. Petersburg Mathematical Journal
SN - 1061-0022
IS - 2
ER -
ID: 121350615