Differential inclusions and Young measures involving prescribed Jacobians

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.

[1]  P. Pedregal.  Parametrized measures and variational principles , 1997 .

[2]  Pablo Pedregal,et al.  Characterizations of young measures generated by gradients , 1991 .

[3]  I. Fonseca,et al.  Weak Continuity and Lower Semicontinuity Results for Determinants , 2005 .

[4]  Yakov Eliashberg,et al.  Introduction to the h-Principle , 2002 .

[5]  J. Moser,et al.  On a partial differential equation involving the Jacobian determinant , 1990 .

[6]  B. Dacorogna,et al.  General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases , 1997 .

[7]  J. Moser.  On the volume elements on a manifold , 1965 .

[8]  Irene Fonseca,et al.  A -Quasiconvexity. lower semicontinuity, and young measures , 1999 .

[9]  Camillo De Lellis,et al.  The $h$-principle and the equations of fluid dynamics , 2011, 1111.2700.

[10]  J. Ball,et al.  Discontinuous equilibrium solutions and cavitation in nonlinear elasticity , 1982, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[11]  M. Gromov,et al.  Partial Differential Relations , 1986 .

[12]  Kari Astala,et al.  Convex integration and the L p theory of elliptic equations. , 2004 .

[13]  S. Muller,et al.  Convex integration for Lipschitz mappings and counterexamples to regularity , 2003 .

[14]  Irene Fonseca,et al.  Analysis of Concentration and Oscillation Effects Generated by Gradients , 1998 .

[15]  B. Dacorogna,et al.  Calculus of Variations On the equation det ∇ u = f with no sign hypothesis , 2009 .

[16]  S. Müller.  Variational models for microstructure and phase transitions , 1999 .

[17]  V. Sverák.  Regularity properties of deformations with finite energy , 1988 .

[18]  Stefan Müller,et al.  Calculus of Variations and Geometric Evolution Problems , 1999 .

[19]  J. Ball,et al.  W1,p-quasiconvexity and variational problems for multiple integrals , 1984 .

[20]  Pablo Pedregal,et al.  Gradient Young measures generated by sequences in Sobolev spaces , 1994 .

[21]  Konstantinos Koumatos,et al.  Orientation-preserving Young measures , 2013, 1307.1007.

[22]  Antje Baer,et al.  Direct Methods In The Calculus Of Variations , 2016 .