Space-time integral currents of bounded variation

Motivated by a recent model for elasto-plastic evolutions that are driven by the flow of dislocations, this work develops a theory of space-time integral currents with bounded variation in time, which enables a natural variational approach to the analysis of rate-independent geometric evolutions. Based on this, we further introduce the notion of Lipschitz deformation distance between integral currents, which arises physically as a (simplified) dissipation distance. Several results are obtained: A Helly-type compactness theorem, a deformation theorem, an isoperimetric inequality, and the equivalence of the convergence in deformation distance with the classical notion of weak* (or flat) convergence. Finally, we prove that the Lipschitz deformation distance agrees with the (integral) homogeneous Whitney flat metric for boundaryless currents. Physically, this means that two seemingly different ways to measure the dissipation actually coincide.

Automated summary

Motivated by a recent model for elasto-plastic evolutions, this work develops a theory of space-time integral currents with bounded variation in time and introduces the notion of Lipschitz deformation distance between integral currents. Several results are obtained, including a compactness theorem, a deformation theorem, an isoperimetric inequality, and the equivalence of convergence in deformation distance with weak* convergence. Finally, it is proven that the Lipschitz deformation distance coincides with the homogeneous Whitney flat metric for boundaryless currents, indicating the equivalence of two seemingly different ways to measure dissipation.