Volume 25, 2019
|Number of page(s)||45|
|Published online||05 November 2019|
A gradient system with a wiggly energy and relaxed EDP-convergence*
Freiburg im Breisgau, Germany.
2 WIAS, Berlin, Germany.
3 WIAS Berlin and Humboldt-Universität zu Berlin, Berlin, Germany.
** Corresponding author: firstname.lastname@example.org
Accepted: 28 October 2018
For gradient systems depending on a microstructure, it is desirable to derive a macroscopic gradient structure describing the effective behavior of the microscopic scale on the macroscopic evolution. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. This new notion generalizes the concept of EDP-convergence, which was introduced in , and is now called relaxed EDP-convergence. Both notions are based on De Giorgi’s energy-dissipation principle (EDP), however the special structure of the dissipation functional in terms of the primal and dual dissipation potential is, in general, not preserved under Gamma-convergence. By using suitable tiltings we study the kinetic relation directly and, thus, are able to derive a unique macroscopic dissipation potential. The wiggly-energy model of Abeyaratne-Chu-James (1996) serves as a prototypical example where this nontrivial limit passage can be fully analyzed.
Mathematics Subject Classification: 35K55 / 35B27 / 35A15 / 49S05 / 49J40 / 49J45
Key words: Variational evolution / energy functional / dissipation potential / gradient flows / Gamma convergence / EDP-convergence / energy-dissipation / balance / homogenization
© EDP Sciences, SMAI 2019
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