Multiscale Problems in Three Applications - Abstract

Mielke, Alexander

Weak convergence and homogenization in Hamiltonian systems

We present a general theory of weak convergence in the sense of Gamma limits for Lagrangian and Hamiltonian systems based on joint recovery operators. We derive conditions that guarantee that the limits of solutions of the parameter-dependent Hamiltonian systems solve the Hamiltonian system associated with the limiting Hamiltonian obtained independently via Gamma convergence. We apply this theory to discrete, polyatomic oscillator chains and show that the limiting behavior is governed by a semilinear wave equation. The convergence in our rather weak setting allows for the treatment of general coefficients of $mathrm L^infty$ type, i.e. coefficients with jumps in the effective equations. This means that reflection and transmission at interfaces between different crystallographic phases is represented correctly by the weak solutions of the effective wave equation with nonsmooth coefficients.