Homogenization Estimates for Singularly Perturbed Operators
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Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
HOMOGENIZATION ESTIMATES FOR SINGULARLY PERTURBED OPERATORS S. E. Pastukhova MIREA, Russian Technological University 78, pr. Vernadskogo, Moscow 119454, Russia [email protected]
UDC 517.946
We study homogenization of a singularly perturbed second order elliptic operator Aε acting in the whole space Rd with ε-periodic coefficients as ε → 0. Perturbation is generated by a fourth order elliptic operator, for example, by the bi-Laplacian Δ2 with a small parameter ε2 . We find approximations of the resolvent (Lε + 1)−1 of order ε and ε2 in different operator norms. Bibliography: 28 titles.
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Introduction
1.1. Historical remarks. Singularly perturbed differential operators are widely used in applications, for example, in the theory of thin elastic plates and shells (cf., for example, [1]). The G-convergence and homogenization issues attracted a lot of attention since the appearance of homogenization theory (cf. [2]–[4]). For an example we can consider the operator acting in Rd (d 2) as follows: (1.1) Lε = ε2 Δ2 + Aε , Aε = − div (aε (x)∇), where Δ is the Laplace operator, ε is a small positive parameter, aε (x) = a(x/ε) and a(x) = {ajk (x)}dj,k=1 is a measurable symmetric periodic matrix with real entries, and for the periodicity cell the unit cube 2 = [−1/2, 1/2)d is taken. It is assumed that the coefficient matrix a satisfies the ellipticity and boundedness conditions ∃λ > 0 :
λ|ξ|2 aξ · ξ λ−1 |ξ|2
∀ξ ∈ Rd .
(1.2)
In the case d = 2, operators of the form (1.1) appear in the theory of strongly bent thin elastic plates for describing a transverse displacement (or a bend) of a stressed thin plate made from a composite material with ε-periodic structure, and the plate thickness is of the same order ε. The first result on homogenization of the Dirichlet boundary value problem for the equation Lε uε = f in a bounded domain Ω was obtained in [2], where the method of compensated compactness is used to prove the convergence of the solution to this problem in L2 (Ω) to the solution to the Dirichlet problem in Ω for the limit (or homogenized) operator of the second Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 149-168. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0724
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order L0 = − div (a0 ∇).
(1.3)
As usual, the homogenized matrix a0 > 0 is constant and is expressed in terms of the solution to the problem on the periodicity cell (cf. (2.12) below). The feature of the homogenization procedure in this case is that the problem on the cell (cf. (2.7) below) includes a fourth order operator i.e., an operator of the same order as the original one, whereas the homogenized operator is of the second order. From the physical point of view this result can be interpreted as the limit passage from an inhomogeneous thin plate to a homogeneous membrane whose effective characteristics preserve the memory of its origin from the thin plate. The operator Lε with lower order terms independent of the parameter ε has a longer history o
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