Γ-limits and relaxations for rate-independent evolutionary problems
- Mielke, Alexander
- Roubíček, Tomáš
- Stefanelli, Ulisse
2010 Mathematics Subject Classification
- 49J40 49S05 35K90
- Rate-independent problems, energetic formulation, Gamma convergence, relaxation, time-incremental minimization, joint recovery sequence
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.
- Calc. Var. Partial Differ. Equ., 31 (2008) pp. 387--416.