WIAS Preprint No. 1078, (2005)

Existence and uniqueness results for general rate-independent hysteresis problems


  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Rossi, Riccarda
    ORCID: 0000-0002-7808-0261

2010 Mathematics Subject Classification

  • 35K85 35K90 49J53 49J40 49S05


  • Rate independent systems, doubly nonlinear equation, subdifferential inclusion, incremental problems, uniform convexity




We consider the special case of doubly nonlinear differential inclusions which are rate independent. The new feature is that the dissipation potential depends not only on the rate but also on the state itself. The energy potential is assumed to be uniformly convex. This corresponds to evolutionary quasivariational inequalities where the constraint set depends on the state itself. A priori estimates are obtained using a special convexity condition for the sum of the energy potential and the directional derivative of the dissipation potential. Using this, an existence result is derived under the additional assumption that the dissipation potential satisfies certain weak continuity properties. Our uniqueness result generalizes previous of [MT04,BKS04] relies on differentiability conditions and a one-sided Lipschitz estimate, also called structure condition in [MT04].

Appeared in

  • Math. Models Methods Appl. Sci., 17 (2007) pp. 81--123.

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