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: email@example.com
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.