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Γ-limits and relaxations for rate-independent evolutionary problems

  • Alexander MielkeEmail author
  • Tomáš Roubíček
  • Ulisse Stefanelli
Article

Abstract

This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals \({\mathcal{E}}\) and the dissipation distance \({\mathcal{D}}\) . For sequences \(({\mathcal{E}}_k)_{k\in {\mathbb{N}}}\) and \(({\mathcal{D}}_k)_{k\in {\mathbb{N}}}\) we address the question under which conditions the limits q of solutions \(q_k : [0, T]\to {\mathcal{Q}}\) satisfy a suitable limit problem with limit functionals \({\mathcal{E}}_\infty\) and \({\mathcal{D}}_\infty\) , which are the corresponding Γ-limits. We derive a sufficient condition, called conditional 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 joint 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 converge 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.

Mathematics Subject Classification (2007)

49J40 49S05 35K90 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Alexander Mielke
    • 1
    • 2
    Email author
  • Tomáš Roubíček
    • 3
    • 4
  • Ulisse Stefanelli
    • 5
  1. 1.Weierstraß-Institut für Angewandte Analysis und StochastikBerlinGermany
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  3. 3.Mathematical InstituteCharles UniversityPraha 8Czech Republic
  4. 4.Institute of Information Theory and AutomationAcademy of SciencesPraha 8Czech Republic
  5. 5.Istituto di Matematica Applicata e Tecnologie Informatiche-CNRPaviaItaly

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