Differential inclusions and young measures involving prescribed Jacobians

Koumatos K, Rindler F, Wiedemann E (2015)

Publication Type: Journal article

Publication year: 2015

Journal

SIAM Journal on Mathematical Analysis Society for Industrial and Applied Mathematics

Book Volume: 47

Pages Range: 1169-1195

Journal Issue: 2

DOI: 10.1137/140968860

Abstract

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 W1,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, W1,p-quasi-convexity and W1,p-orientation-preserving quasi-convexity are both unsuitable convexity conditions for nonlinear elasticity where the energy is assumed to blow up as the Jacobian approaches zero.

Authors with CRIS profile

Emil Wiedemann

Involved external institutions

Gran Sasso Science Institute IT Italy (IT) University of Warwick GB United Kingdom (GB) Rheinische Friedrich-Wilhelms-Universität Bonn DE Germany (DE)

How to cite

APA:

Koumatos, K., Rindler, F., & Wiedemann, E. (2015). Differential inclusions and young measures involving prescribed Jacobians. SIAM Journal on Mathematical Analysis, 47(2), 1169-1195. https://dx.doi.org/10.1137/140968860

MLA:

Koumatos, Konstantinos, Filip Rindler, and Emil Wiedemann. "Differential inclusions and young measures involving prescribed Jacobians." SIAM Journal on Mathematical Analysis 47.2 (2015): 1169-1195.

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