Regularity of stresses in Prandtl-Reuss perfect plasticity

We study the differential properties of solutions of the Prandtl-Reuss model. We prove that in dimensions n = 2, 3 the stress tensor has locally square-integrable first derivatives: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma\in L^\infty([0,T];W^{1,2}_{{\rm loc}}(\Omega;{\mathbb M}^{n{\times}n}_{{\rm sym}}))$$\end{document} . The result is based on discretization of time and uniform estimates of solutions of the incremental problems, which generalize the estimates in the case of Hencky perfect plasticity. Counterexamples to the regularity of displacements and plastic strains in the quasistatic case are presented.