Calculus of Variations and Geometric Measure Theory

A. Braides - G. A. Francfort

Bounds on the effective behavior of a square conducting lattice

created on 29 Mar 2003
modified by braidesa on 08 Feb 2005


Published Paper

Inserted: 29 mar 2003
Last Updated: 8 feb 2005

Journal: Proc. R. Soc. London A.
Volume: 460
Pages: 1755-1769
Year: 2004


A collection of resistors with two possible resistivities is considered. This paper investigates the overall or macroscopic behavior of a square two--dimensional lattice of such resistors when each type is present in fixed proportion in the lattice. The macroscopic behavior is that of an anisotropic conductor at the continuum level and the goal of the paper is to describe the set of all possible such conductors.This is thus a problem of bounds in the footstep of an abundant literature on the topic in the continuum case. The originality of the paper is that the investigation focusses on the interplay between homogenization and the passage from a discrete network to a continuum. A set of bounds is proposed and its optimality is shown when the proportion of each resistor on the discrete lattice is 12. We conjecture that the derived bounds are optimal for all proportions.