5 Mar 2009; Mat-Nat Faculty (UiTø), room A228. Thursday 14:15-16:00

Patrizia Donato (University of Rouen and University Paris 6, France)

"Homogenization of a linear hyperbolic problem with a memory effect"

The aim of the homogenization theory is to describe the macroscopic properties (modeling for instance the heat diffusion, wave propagation etc) of composite materials which are a fine mixture of two or more constituents.

Mathematically, one has to study the asymptotic behavior of a PDE with oscillating coefficients, depending on a small parameter which converges to zero  and describes the heterogeneities. The domain can also depend on the small parameter, like in the case of perforated domains or thin structures. The aim is to prove that the solution converges to that of the PDE (called the homogenized problem), explicitly described, which gives the macroscopic behavior of the material.

In the periodic case, the heterogeneities are periodically distributed, and the small parameter represents the period. The homogenization methods, reducing the microscopic analysis to some cell problems, rend the numerical computations more accessible.

The talk will focus on some homogenization results for the wave equation with rapidly oscillating coefficients in a periodic two-component composite with imperfect inclusions. We prescribe on the interface between the two components a jump of the solution proportional to the conormal derivatives with a coefficient being a function of the small parameter. This condition describes an imperfect contact. For the different convergence rates of the proportionality function, we obtain different limit problems. In the most interesting case, we have a memory effect  in the homogenized problem.

Some additional convergence results (correctors) will also be presented.