Spectral approach to homogenization of an elliptic operator periodic in some directions
✍ Scribed by R. Bunoiu; G. Cardone; T. Suslina
- Publisher
- John Wiley and Sons
- Year
- 2011
- Tongue
- English
- Weight
- 334 KB
- Volume
- 34
- Category
- Article
- ISSN
- 0170-4214
- DOI
- 10.1002/mma.1424
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✦ Synopsis
The operator
A e = D 1 g 1 (x 1 / e,x 2 )D 1 +D 2 g 2 (x 1 / e,x 2 )D 2 is considered in L 2 (R 2 ), where g j (x 1 ,x 2 ), j = 1, 2, are periodic in x 1 with period 1, bounded and positive definite. Let function Q(x 1 ,x 2 ) be bounded, positive definite and periodic in x 1 with period 1. Let Q e (x 1 ,x 2 ) = Q(x 1 / e,x 2 ). The behavior of the operator (A e +Q e ) -1 as e → 0 is studied. It is proved that the operator (A e +Q e ) -1 tends to (A 0 +Q 0 ) -1 in the operator norm in L 2 (R 2 ). Here, A 0 is the effective operator whose coefficients depend only on x 2 , Q 0 is the mean value of Q in x 1 . A sharp order estimate for the norm of the difference (A e +Q e ) -1 -(A 0 +Q 0 ) -1 is obtained. The result is applied to homogenization of the Schrödinger operator with a singular potential periodic in one direction.