WIAS Preprint No. 1156, (2006)

Γ-limits and relaxations for rate-independent evolutionary problems



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Roubíček, Tomáš
  • Stefanelli, Ulisse

2010 Mathematics Subject Classification

  • 49J40 49S05 35K90

Keywords

  • Rate-independent problems, energetic formulation, Gamma convergence, relaxation, time-incremental minimization, joint recovery sequence

DOI

10.20347/WIAS.PREPRINT.1156

Abstract

This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals ε and the dissipation distance D. For sequences (ε k)k ∈ ℕ and (D k)k ∈ ℕ we address the question under which conditions the limits q of solutions qk: [0,T] → Q satisfy a suitable limit problem with limit functionals ε and D, which are the corresponding Γ-limits. We derive a sufficient condition, called emphconditional upper semi-continuity of the stable sets, which is essential to guarantee that q solves the limit problem. In particular, this condition holds if certain emphjoint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator convergece if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model.

Appeared in

  • Calc. Var. Partial Differ. Equ., 31 (2008) pp. 387--416.

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