Alexander Mielke

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This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The(More)
We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers(More)
This paper is devoted to the homogenization for a class of rate-independent systems described by the energetic formulation. The associated nonlinear partial differential system has periodically oscillating coefficients, but has the form of a standard evolutionary variational inequality. Thus, the model applies to standard linearized elastoplasticity with(More)
We consider rate-independent models which are defined via two functionals: the time-dependent energy-storage functional I : [0, T ]×X → [0,∞] and the dissipation distance D : X × X → [0,∞]. A function z : [0, T ] → X is called a solution of the energetic model, if for all 0 ≤ s < t ≤ T we have stability: I(t, z(t)) ≤ I(t, z̃) +D(z(t), z̃) for all z̃ ∈ X;(More)
We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith’s local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this(More)
We are interested in the long{time behavior of nonlinear parabolic PDEs deened on unbounded cylindrical domains. For dissipative systems deened on bounded domains, the long{time behavior can often be described by the dynamics in their nite{dimensional attractors. For systems deened on the innnite line, very little is known at present, since the lack of(More)
Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended(More)
We consider the dynamics of infinite harmonic lattices in the limit of the lattice distance ε tending to 0. We allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by Fourier transform and linear algebra. Our aim is to derive macroscopic continuum limit equations for ε → 0. For the weak limit of(More)