OPTIMAL BOUNDS ON DISPERSION COEFFICIENT IN ONE-DIMENSIONAL PERIODIC MEDIA CARLOS CONCA * and JORGE SAN MARTÍN y Departamento de Ingenier {a Matem atica, Facultad de Ciencias F {sicas y Matem aticas, Universidad de Chile and Centro de Modelamiento Matem atico, UMR 2071 CNRS — UChile, Casilla 170/3Correo 3, Santiago, Chile * cconca@dim.uchile.cl y jorge@dim.uchile.cl LOREDANA SMARANDA z Department of Mathematics, Faculty of Mathematics and Computer Science, University of Pite» sti, 110040 Pite» sti, Str. T ^ argu din Vale, nr. 1, Romania smaranda@dim.uchile.cl MUTHUSAMY VANNINATHAN TIFR-CAM, Post Bag 6503, GKVK Post, Bangalore 560065, India vanni@math.tifrbng.res.in Received 21 January 2009 Communicated by F. Brezzi In this paper, we consider the macroscopic quantity, namely the dispersion tensor associated with a periodic structure in one dimension (see Refs. 5 and 7). We describe the set in which this quantity lies, as the microstructure varies preserving the volume fraction. Keywords: Homogenization; homogenized matrix; dispersion tensor; periodic media; bounds. AMS Subject Classi¯cation: 35B27, 74Q20, 78M40 1. Introduction This work is about macroscopic behavior of ¯ne periodic structures with small period denoted by ". It is well-known that (see Ref. 3) the homogenization of these struc- tures leads to the ¯rst macroscopic quantity, namely the homogenized matrix q ‡ Corresponding author Mathematical Models and Methods in Applied Sciences Vol. 19, No. 9 (2009) 17431764 # . c World Scienti¯c Publishing Company Doi: 10.1142/S0218202509003930 1743 Math. Models Methods Appl. Sci. 2009.19:1743-1764. Downloaded from www.worldscientific.com by UNIVERSIDAD DE CHILE SISTEMA DE SERVICIOS DE INFORMACION Y BIBLIOTECAS (SISIB) on 03/18/13. For personal use only.