The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems

¹ Weierstrass Institute Berlin, Mohrenstraße 39, 10117 Berlin, Germany.
² Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria.

^* Corresponding author: matthias.liero@wias-berlin.de

Received: 26 June 2017
Accepted: 22 March 2018

Abstract

We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizers correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ → 0, ε → 0, as well as δ + ε → 0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε → 0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization and a dimension reduction problem as examples of application.