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SUMMARY:Effective Properties of Doubly Periodic Media - Richard Craster (I
 mperial College London)
DTSTART:20170303T115500Z
DTEND:20170303T121500Z
UID:TALK71230@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:In his seminal paper "A theorem on the conductivity of a compo
 site medium"\, J. Math. Phys\, 5\, 1964 Joe Keller in just two pages eluci
 dated fundamental properties for the effective conductivities of composite
  media\; the setting was in electrostatics but the application is far broa
 der than this.  This\, together with a parallel Russian literature dominat
 ed by Dykhne&#39\;s 1971 result and others\, has provided the bedrock of m
 uch of the theory of composite media\, at least in the static setting. It 
 seems only fitting that Joe Keller&#39\;s contribution to this area be hig
 hlighted.  <span><br><br>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 t
 urns out to be geometric mean of the conductivities. Surprisingly for doub
 ly periodic tilings of the plane there are few other known exact solutions
 .  One case is that of the four phase checkerboard\, so a square subdivide
 d into four equal squares that are then of different colours\, that then t
 iles the plane\; or more generally rectangles.  In 1985 Mortola and Steffe
  conjectured the result and it was\, for a while a little controversial in
 deed 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\, usi
 ng 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\, K
 eller&#39\;s contribution and some reminiscences of the discussion I had w
 ith him on this. <br><br>This work was co-authored with Yuri Obnosov<br></
 span>
LOCATION:Seminar Room 1\, Newton Institute
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