caa a22 4500 605515883 CHVBK 20210128100711.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s00205-015-0890-x doi (NATIONALLICENCE)springer-10.1007/s00205-015-0890-x Optimal Flux Densities for Linear Mappings and the Multiscale Geometry of Structured Deformations [Elektronische Daten] [David Owen, Roberto Paroni] We establish the unexpected equality of the optimal volume density of total flux of a linear vector field $${x \longmapsto Mx}$$ x ⟼ M x 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 $${\mathcal{S}}$$ S of $${N{\times}N}$$ 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 $${x \longmapsto Mx}$$ x ⟼ M x 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 $${x \longmapsto Mx}$$ x ⟼ M x and the volume fraction swept out by submacroscopic switches and interpenetrations are shown to have the common value (trM)−. Springer-Verlag Berlin Heidelberg, 2015 Owen David Department of Mathematical Sciences, Carnegie Mellon University, 15213, Pittsburgh, PA, USA aut Paroni Roberto DADU, Università degli Studi di Sassari, Palazzo del Pou Salit, Piazza Duomo, 07041, Alghero, Italy aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/3(2015-12-01), 1633-1652 0003-9527 218:3<1633 2015 218 205 https://doi.org/10.1007/s00205-015-0890-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-015-0890-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Owen David Department of Mathematical Sciences, Carnegie Mellon University, 15213, Pittsburgh, PA, USA aut NATIONALLICENCE 700 1- Paroni Roberto DADU, Università degli Studi di Sassari, Palazzo del Pou Salit, Piazza Duomo, 07041, Alghero, Italy aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/3(2015-12-01), 1633-1652 0003-9527 218:3<1633 2015 218 205