Generic Ill-posedness of the Energy-Momentum Equations and Differential Inclusions

Abstract: We show that the energy-momentum equations arising from inner variations whose Lagrangian satisfies a generic symmetry condition are generically ill-posed. This is done by proving that there exists a subclass of Lipschitz solutions that are also solutions to a differential inclusion. In particular these solutions can be nowhere C1. We prove that these solutions are not stationary points if the Lagrangian W is C1 and strictly rank-one convex. In view of the Lipschitz regularity result of Iwaniec, Kovalev and Onninen for solution of the energy-momentum equation in dimension 2 we give a sufficient condition for the non-existence of a partial C1-regularity result even under the condition that the mappings satisfy a positive Jacobian determinant condition. Finally we consider a number of well-known functionals studied in nonlinear elasticity and geometric function theory and show that these do not satisfy this obstruction to partial regularity.