Bloch wave homogenization of scalar elliptic operators

Article type: Research Article

Authors: Ganesh, S. Sivaji | Vanninathan, M.

Affiliations: IISc‐TIFR Mathematics Programme, TIFR Centre, PO Box 1234, Bangalore, 560012, India E‐mail: siva@math.tifrbng.res.in, vanni@math.tifrbng.res.in

Abstract: Periodic homogenization result for selfadjoint operators via Bloch wave method was obtained by Conca and Vanninathan in [12]. Even though the spectral tools used in [12] are not available in non‐selfadjoint case, it is possible to recover the complete homogenization result of Murat and Tartar in the periodic case through the Bloch wave method. A dominant Bloch mode is introduced and plays the key role in the homogenization process. It is also established that the remainder does not contribute in the homogenization process. This requires separation of scales between the dominant Bloch mode and the rest. This separation is proved via a Poincaré‐type inequality. Further, the proof of homogenization theorem of [12] is simplified.

Keywords: Bloch waves, homogenization

Journal: Asymptotic Analysis, vol. 39, no. 1, pp. 15-44, 2004