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Differential inclusions and young measures involving prescribed Jacobians

journal contribution
posted on 2023-06-09, 04:54 authored by Konstantinos KoumatosKonstantinos Koumatos, Filip Rindler, Emil Wiedemann
This work presents a general principle, in the spirit of convex integration, leading to a method for the characterization of Young measures generated by gradients of maps in W^{1,p} with p less than the space dimension, whose Jacobian determinant is subjected to a range of constraints. Two special cases are particularly important in the theories of elasticity and fluid dynamics: when (a) the generating gradients have positive Jacobians that are uniformly bounded away from zero and (b) the underlying deformations are incompressible, corresponding to their Jacobian determinants being constantly one. This characterization result, along with its various corollaries, underlines the flexibility of the Jacobian determinant in subcritical Sobolev spaces and gives a more systematic and general perspective on previously known pathologies of the pointwise Jacobian. Finally, we show that, for p less than the dimension, W^{1,p}-quasiconvexity and W^{1,p}-orientation-preserving quasiconvexity are both unsuitable convexity conditions for nonlinear elasticity where the energy is assumed to blow up as the Jacobian approaches zero.

History

Publication status

  • Published

File Version

  • Published version

Journal

SIAM Journal on Mathematical Analysis

ISSN

0036-1410

Publisher

Society for Industrial and Applied Mathematics

Issue

2

Volume

47

Page range

1169-1195

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Analysis and Partial Differential Equations Research Group Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2017-01-24

First Compliant Deposit (FCD) Date

2017-01-24

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