In this work in collaboration with Gilles Francfort, we study the spatial hyperbolic structure of a two-dimensional scalar model of perfect plasticity. Minimizing solutions suffer from two pathologies: the linear growth of the energy leads to singularities in the energy space, while the absence of strict convexity yields non uniqueness of the solutions. Taking advantage of the hyperbolic system associated to this model and under the assumption that the plastic zone has non empty interior, we show rigidity properties of the solutions in that set and, in particular, their constancy along characteristic lines. It allows us to precisely describe the geometric structure of the plastic zone and show partial uniqueness results when it touches the boundary of the domain.
