Quasiconvexity and Relaxation in Optimal Transportation of Closed Differential Forms

This manuscript extends the relaxation theory from nonlinear elasticity to electromagnetism and to actions defined on paths of differential forms. The introduction of a gauge allows for a reformulation of the notion of quasiconvexity in Bandyopadhyay et al. (J Eur Math Soc 17:1009-1039, 2015), from the static to the dynamic case. These gauges drastically simplify our analysis. Any non-negative coercive Borel cost function admits a quasiconvex envelope for which a representation formula is provided. The action induced by the envelope not only has the same infimum as the original action, but has the virtue to admit minimizers. This completes our relaxation theory program.