WIAS Preprint No. 2411, (2017)

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



Authors

  • Liero, Matthias
    ORCID: 0000-0002-0963-2915
  • Melchionna, Stefano

2010 Mathematics Subject Classification

  • 58E30 35K55 47J35

Keywords

  • Doubly nonlinear evolution, weighted-energy-dissipation principle, evolutionary Gamma-convergence, variational principle

DOI

10.20347/WIAS.PREPRINT.2411

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 minimizer 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 problem as an example of application.

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