Optimal flux densities for linear mappings and the multiscale geometry of structured deformations

We establish the unexpected equality of the optimal volume density of total flux of a linear vector field x 7−→ Mx and the least volume fraction that can be swept out by submacroscopic switches, separations, and interpenetrations associated with the purely submacroscopic structured deformation (i, I +M). This equality is established first by identifying a dense set S of N×N matrices M for which the optimal total flux density equals |trM |, the absolute value of the trace of M . We then use known representation formulae for relaxed energies for structured deformations to show that the desired least volume fraction associated with (i, I + M) also equals |trM |. We also refine the above result by showing the equality of the optimal volume density of the positive part of the flux of x 7−→ Mx and the volume fraction swept out by submacroscopic separations alone, with common value (trM)+. Similarly, the optimal volume density of the negative part of the flux of x 7−→Mx and the volume fraction swept out by submacroscopic switches and interpenetrations are shown to have the common value (trM)−.