Homogenization of a class of linear partial differential equations

摘要

We present a unified Hilbert space perspective to homogenization of a class of evolutionary equations of mathematical physics. We formulate homogenization in a purely operator-theoretic setting. Using "A structural observation for linear material laws in classical mathematical physics" by R. Picard [Mathematical Methods in the Applied Sciences 32 (2009), 1768-1803], we discuss constitutive relations as certain elements of the Hardy space H~∞(E; L(H)) of bounded, analytic and operator-valued functions M :E → L(H), where E C C open, H Hilbert space. The core idea is to introduce a certain topology on the set of constitutive relations. Given a convergent sequence of constitutive relations, the behavior of solutions to the respective problems is discussed. We apply the results to the equations of acoustics, thermodynamics, elasticity or coupled systems such as thermo-elasticity. The respective equations may also incorporate memory or delay terms and fractional derivatives. In particular, constitutive relations via differential equations can also be treated.