Space-time integral currents of bounded variation

TL;DR

This paper develops a theory of space-time integral currents with bounded variation to analyze rate-independent evolutions, introducing a Lipschitz deformation distance that aligns with classical convergence notions and physical dissipation measures.

Contribution

It introduces a novel framework for space-time integral currents with bounded variation and a Lipschitz deformation distance, connecting physical dissipation with mathematical convergence.

Findings

Established a Helly-type compactness theorem.

Proved the equivalence of deformation distance with weak* convergence.

Showed the Lipschitz deformation distance matches the Whitney flat metric.

Abstract

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.