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Publication 14-CNA-020
Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of Structured Deformations David R. Owen Roberto Paroni Abstract: We establish the unexpected equality of the optimal volume density of total flux of a linear vector field $x\longmapsto Mx$ 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 $\mathcal{S}$ of $N\times N$ matrices $M$ for which the optimal total flux density equals $\left\vert trM\right\vert $, 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 $\left\vert trM\right\vert $. \ We also refine the above result by showing the equality of the optimal volume density of the positive part of the flux of $x\longmapsto Mx$ 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 $x\longmapsto Mx$ and the volume fraction swept out by submacroscopic switches and interpenetrations are shown to have the common value $(trM)^{-}$. Get the paper in its entirety as |
