Differential inclusions and young measures involving prescribed Jacobians

Koumatos, Konstantinos, Rindler, Filip and Wiedemann, Emil (2015) Differential inclusions and young measures involving prescribed Jacobians. SIAM Journal on Mathematical Analysis, 47 (2). pp. 1169-1195. ISSN 0036-1410

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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 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Subjects: Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Depositing User: Konstantinos Koumatos
Date Deposited: 24 Jan 2017 15:35
Last Modified: 08 Mar 2017 06:15
URI: http://sro.sussex.ac.uk/id/eprint/66429

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