What is GTH, the general theory of homogenization?

Friday, April 6, 2012 - 16:00 to 16:50
Speaker Information
Luc Tartar
Carnegie Mellon University

Abstract or Additional Information

The mathematical theory of homogenization (started in the late 1960s in Pisa, and in the early 1970s in Paris) is not about studying partial differential equations with periodically modulated coefficients (to which some add a stochastic grain of salt according to their taste), but to define in a mathematical way the properties of mixtures: for some systems of partial differential equations, preferably resembling some which have a physical interpretation in continuum mechanics or physics, one considers a sequence of coefficients converging weakly but not strongly, so that it permits to consider a fine mixture of various materials, and one tries to understand which adapted topologies (usually of various weak types) should be used for the coefficients and the solutions in order to define an effective equation governing the mixture, which may have a much more general form than in the case of clear interfaces between distinct materials.

For some elliptic equations of variational type, it looks like a nonlinear microlocal theory, but for more interesting examples of hyperbolic type, which are far from being completely understood, the limiting equations are not always partial differential equations.

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