TY - JOUR

T1 - Homogenization for Locally Periodic Elliptic Problems on a Domain

AU - Сеник, Никита Николаевич

PY - 2023/4/26

Y1 - 2023/4/26

N2 - Let Ω be a Lipschitz domain in Rd, and let A∊ = - div A(x, x/∊)∇ be a strongly elliptic operator on Ω. We suppose that ∊ is small and the function A is Hölder continuous of order s ∊ (0, 1] in the first variable and periodic in the second, so the coefficients of A∊ are locally periodic and rapidly oscillate. Given μ in the resolvent set, we are interested in finding the rates of approximations, as ∊ → 0, for (A∊ - μ)-1 and ∇(A∊ - μ)-1 in the operator topology on Lp for suitable p. It is well-known that the rates depend on regularity of the effective operator A0. We prove that if (A0 - μ)-1 and its adjoint are bounded from Lp(Ω)n to the Lipschitz-Besov space Λ1+sp(Ω)n with the same s as for A, then the rates are, respectively, ∊s and ∊s/p. The results are applied to the Dirichlet, Neumann, and mixed Dirichlet-Neumann problems for strongly elliptic operators with uniformly bounded and vanishing mean oscillation coefficients.

AB - Let Ω be a Lipschitz domain in Rd, and let A∊ = - div A(x, x/∊)∇ be a strongly elliptic operator on Ω. We suppose that ∊ is small and the function A is Hölder continuous of order s ∊ (0, 1] in the first variable and periodic in the second, so the coefficients of A∊ are locally periodic and rapidly oscillate. Given μ in the resolvent set, we are interested in finding the rates of approximations, as ∊ → 0, for (A∊ - μ)-1 and ∇(A∊ - μ)-1 in the operator topology on Lp for suitable p. It is well-known that the rates depend on regularity of the effective operator A0. We prove that if (A0 - μ)-1 and its adjoint are bounded from Lp(Ω)n to the Lipschitz-Besov space Λ1+sp(Ω)n with the same s as for A, then the rates are, respectively, ∊s and ∊s/p. The results are applied to the Dirichlet, Neumann, and mixed Dirichlet-Neumann problems for strongly elliptic operators with uniformly bounded and vanishing mean oscillation coefficients.

KW - corrector

KW - effective operator

KW - homogenization

KW - locally periodic operators

KW - operator error estimates

UR - https://epubs.siam.org/doi/full/10.1137/21M1419337

UR - https://www.mendeley.com/catalogue/8e5ccbc3-0e44-3634-bdab-6ee958c08dbf/

U2 - 10.1137/21M1419337

DO - 10.1137/21M1419337

M3 - Article

VL - 55

SP - 849

EP - 881

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -