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Effective Properties of Doubly Periodic Media

Presented by: 
Richard Craster Imperial College London
Friday 3rd March 2017 - 11:55 to 12:15
INI Seminar Room 1
In his seminal paper "A theorem on the conductivity of a composite medium", J. Math. Phys, 5, 1964 Joe Keller in just two pages elucidated fundamental properties for the effective conductivities of composite media; the setting was in electrostatics but the application is far broader than this. This, together with a parallel Russian literature dominated by Dykhne's 1971 result and others, has provided the bedrock of much of the theory of composite media, at least in the static setting. It seems only fitting that Joe Keller's contribution to this area be highlighted.

A remarkably simple result is that of the square infinite checkerboard, so a plane compose of black and white squares with each phase having a different conductivity; the effective conductivity turns out to be geometric mean of the conductivities. Surprisingly for doubly periodic tilings of the plane there are few other known exact solutions. One case is that of the four phase checkerboard, so a square subdivided into four equal squares that are then of different colours, that then tiles the plane; or more generally rectangles. In 1985 Mortola and Steffe conjectured the result and it was, for a while a little controversial indeed it was alleged to be wrong by, I think, Kozlov who promptly passed away without saying why. In 2001 myself and Obnosov proved the result and simultaneously using a very different approach by Milton. This talk, using the same overheads as I used in a talk attended by Joe Keller in 2000, will describe the area and hopefully provide some insight to the area, Keller's contribution and some reminiscences of the discussion I had with him on this.

This work was co-authored with Yuri Obnosov
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons