WIAS Preprint No. 1219, (2007)

Weak-convergence methods for Hamiltonian multiscale problems



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 35B27 74Q10 37Kxx 37K60 70Hxx

Keywords

  • Homogenization, infinite-dimensional Hamiltonian and Lagrangian, effective Hamiltonian, wave equation, oscillator chain, Gamma convergence, recovery operators

Abstract

We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $rmL^infty$ coefficients

Appeared in

  • Discrete Contin. Dyn. Syst., 20 (2008) pp. 53--79.

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