On the strict approximation of currents and space-time trajectories

Abstract

This work establishes a wide variety of strict approximation results for space-time trajectories, a class of space-time currents that are used to model the evolution of dislocations in crystalline materials. In particular, we introduce a strict-type notion of convergence which respects quantities associated with geometric transport phenomena and prove approximation results in this topology. These results build on the classical strict approximation results for normal and integral currents developed by Federer and Fleming, which we also review and refine.