# Analysis Seminar

#### Labile 2-dimensional ether through homogenization

Speaker: Gilles FRANCFORT, Universite Paris XIII

Location: Warren Weaver Hall 1302

Date: Thursday, September 20, 2018, 11 a.m.

Synopsis:

Homogenization in linear elliptic problems usually assumes
coercivity of the accompanying Dirichlet form.  In contrast with the
scalar case, coercivity in linear elasticity is not ensured through mere
(strong) ellipticity and a stronger notion of very strong ellipticity is
usually assumed to hold.

Yet a homogenization process can still be performed, very strong ellipticity
notwithstanding, for a class of two- phase mixtures giving rise to an
overall behavior for which strict ellipticity can be lost.

That result is at the root of the construction of a two-dimensional medium
which can propagate plane waves in a bounded  domain  with  Dirichlet
boundary  conditions,  a  possibility  which  does not  exist  for  the
associated  two-phase micro-structure at a fixed scale.

Equally striking is the realization that such a material blocks
longitudinal waves in the direction of lamination, thereby acting as some
kind of two-dimensional aether in the sense of e.g.  Cauchy or Maxwell.