Γ-limits and relaxations for rate-independent evolutionary problems

• DOI: 10.1007/s00526-007-0119-4
• Publication Date: 2007-07-16T21:56:31Z
• Authors: Alexander Mielke; Tomáš Roubíček; Ulisse Stefanelli
• Keywords: ddc:510; 35K90; article; energetic formulation; 49S05; 01 natural sciences; 510; joint recovery sequence; relaxation; Rate-independent problems -- energetic formulation -- Gamma convergence -- relaxation -- time-incremental minimization -- joint recovery sequence; Gamma convergence; 0101 mathematics; 49J40; time-incremental minimization; Rate-independent problems

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.