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Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of Structured Deformations

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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 |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 \({x \longmapsto Mx}\) 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)−.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    David R. Owen

  2. DADU, Università degli Studi di Sassari, Palazzo del Pou Salit, Piazza Duomo, 07041, Alghero, Italy

    Roberto Paroni

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  1. David R. Owen
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  2. Roberto Paroni
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Correspondence to Roberto Paroni.

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Communicated by C. Dafermos

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Owen, D.R., Paroni, R. Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of Structured Deformations. Arch Rational Mech Anal 218, 1633–1652 (2015). https://doi.org/10.1007/s00205-015-0890-x

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  • DOI: https://doi.org/10.1007/s00205-015-0890-x

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