On Homogenization of Locally Periodic Elliptic and Parabolic Operators
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Homogenization of Locally Periodic Elliptic and Parabolic Operators N. N. Senik Received May 13, 2019; in final form, June 13, 2019; accepted June 15, 2019
Abstract. Let Ω be a C 1,s bounded domain (s > 1/2) in Rd , and let Aε = − div A(x, x/ε)∇ be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of Aε are locally periodic. For μ in the resolvent set, we are interested in finding the rates of approximations, as ε → 0, for (Aε − μρε )−1 and ∇(Aε − μρε )−1 in the operator topology on L2 . Here ρε (x) = ρ(x, x/ε) is a positive definite locally periodic function with ρ satisfying the same assumptions as A. Keeping track of the rate dependence on both ε and μ, we then proceed to similar questions for the solution to the initial boundary-value problem ρε ∂t vε = −Aε vε . Key words: homogenization, operator error estimates, locally periodic operators, elliptic systems, parabolic systems. DOI: 10.1134/S0016266320010104
1. Problem formulation. Let Ω be a bounded domain in Rd with ∂Ω ∈ C 1,s , where s > ¯ L ˜ ∞ (Q))n×n be 1/2, and let Q be the unit cube in Rd centered at the origin. Let Akl ∈ C 0,1 (Ω; d ¯ complex-valued mappings on Ω × R that are Lipschitz in the first variable and periodic (with respect to the lattice Zd ) in the second one. We set D = −i∇ and Aεkl (x) = Akl (x, x/ε), ε > 0, and consider the matrix operator d A ε = D ∗ Aε D = Dk Aεkl Dl (1) k,l=1
˚1 (Ω)n of functions with zero trace on ∂Ω to its dual H −1 (Ω)n . from the complex Sobolev space H If ε is small, then the coefficients of A ε rapidly oscillate with, roughly speaking, slowly varying amplitude and are therefore called locally periodic. Notice that the regularity condition on the function A = {Akl }dk,l=1 in one of the variables only ensures that the amplitude of Aε = {Aεkl }dk,l=1 changes smoothly, still Aε itself may have jumps. We assume that the operator A ε is strongly elliptic and, moreover, strongly coercive uniformly in ε in an interval E = (0, ε0 ]; that is, for any ε ∈ E, Re(Aε Du, Du)L2 (Ω) cA Du2L2 (Ω) ,
˚1 (Ω)n , u∈H
(2)
with cA > 0. Then, by the Friedrichs inequality, we also have Re(Aε Du, Du)L2 (Ω) cA CF−2 u2L2 (Ω) ,
˚1 (Ω)n , u∈H
(3)
where CF is the constant in the Friedrichs inequality (for example, one may take CF = diam Ω). It follows that, for all ε ∈ E, the spectrum of A ε is contained in the truncated sector −2 (4) SI = z ∈ C : |Im z| c−1 A AL∞ Re z and Re z cA CF and that A ε itself is m-sectorial. Now we define the family of truncated sectors S(δ, γ) = z ∈ C : |Im z| δ Re z and Re z γ . 68
c 2020 by Pleiades Publishing, Ltd. 0016–2663/19/5303–0174
(5)
The sector SI belongs to this family; it corresponds to the parameters δI = c−1 A AL∞ and γI = ¯ L ˜ ∞ (Q))n×n be a function such that the matrix ρ(x, y) is self-adjoint cA CF−2 . Next, let ρ ∈ C 0,1 (Ω; and positive definite uniformly in x and y. By Sρ we denote the truncated sector (5) with δ
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