L 2 - Approximation of Resolvents in Homogenization of Higher Order Elliptic Operators
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Journal of Mathematical Sciences, Vol. 251, No. 6, December, 2020
L2 - APPROXIMATION OF RESOLVENTS IN HOMOGENIZATION OF HIGHER ORDER ELLIPTIC OPERATORS S. E. Pastukhova MIREA, Russian Technological University 78, pr. Vernadskogo, Moscow 119454, Russia [email protected]
UDC 517.946
We study homogenization of a divergence form elliptic operator Aε of order 2m 4 with is of ε-periodic coefficients, where ε is a small parameter. The homogenized operator A the same type as Aε , but with constant coefficients. For selfadjoint operators Aε without lower order terms we obtain an estimate of order ε2 for the difference of resolvents + 1)−1 in the operator (L2 →L2 )–norm. In the non-selfadjoint case, (Aε + 1)−1 and (A + 1)−1 + εK1 + O(ε2 ) with corrector K1 . To we find an approximation (Aε + 1)−1 = (A prove the operator estimate, we use the shift method. Bibliography: 14 titles.
1
Introduction
The operator homogenization estimates were mainly considered for linear second order differential operators with periodic coefficients, whereas higher order operators have not been studied in this regard until recently. The present paper is devoted to homogenization for higher order operators. In this paper, we continue the study of [1] and use the approach proposed in [2]. 1.1. We consider the following equation of an even order 2m4 in the whole space Rd : uε ∈ H m (Rd ), Aε = (−1)m
(Aε + 1)uε = f, f ∈ L2 (Rd ), D α (aεαβ (x)D β ),
(1.1)
|α|=|β|=m
with rapidly oscillating coefficients aεαβ (x) = aαβ (y)|y=ε−1 x for small ε ∈ (0, 1). Here, D α denotes the multiderivative ∂ , i = 1, . . . , d, D α = D1α1 . . . Ddαd , Di = ∂xi where α = (α1 , . . . , αd ) is the multiindex of length |α| = α1 + . . . + αd with αj ∈ Z0 , the coefficients aαβ (y) are measurable periodic real-valued functions, and Y = [−1/2, 1/2)d is the periodicity cell. Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 113-132. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2516-0902
902
We assume that the following symmetry, boundedness, and ellipticity conditions hold: aαβ = aβα ,
aαβ L∞ (Y ) λ1 , aαβ (x)D β ϕD α ϕ dx λ0
Rd |α|=|β|=m
|D α ϕ|2 dx ∀ϕ ∈ C0∞ (Rd )
(1.2)
Rd |α|=m
for some positive λ0 , λ1 and all multiindices α and β of length m. In (1.1), H m = H m (Rd ) is the Sobolev space equipped with the norm |D α u|2 dx. u2H m = Rd |α|m
As known, the set of smooth compactly functions is dense in H m (Rd ) and the norm can be equivalently introduced by 2 α 2 |D u| dx + |u|2 dx, uH m = Rd |α|=m
Rd
The G-convergence and homogenization issues for differential operators Aε in (1.1) has been studied since the 70s. For example, the divergence form operators (−1)|α| D α (aεαβ (x)D β ) (1.3) Aε = |α|m,|β|m
with lower order terms are considered in [3], where the symmetry condition in (1.2) and the periodicity of coefficients are not required. The well-known result on homogenization of the operator Aε in (1.1) means the closeness in the sense of the strong operator topology between + 1)−1 of the the
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