A gradient system with a wiggly energy and relaxed EDP-convergence^*

¹ Universität Freiburg, Freiburg im Breisgau, Germany.
² WIAS, Berlin, Germany.
³ WIAS Berlin and Humboldt-Universität zu Berlin, Berlin, Germany.

^** Corresponding author: mielke@wias-berlin.de

Received: 13 January 2018
Accepted: 28 October 2018

Abstract

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 [26], 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.