In this work, in collaboration with Gilles Francfort, we are interested in the hyperbolic structure with respect to the spatial variables of a variational problem arising in the theory of plasticity. Minimizing solutions suffer of two pathologies: the absence of strict convexity of the energy leads to nonuniqueness of minimizers, and the linear growth at infinity implies the appearance of singularities. Taking care of the hyperbolic character of the underlying system of PDEs, the analysis of the characteristic flow leads to rigidity properties of the solutions. It allows one to describe accurately the geometric structure of the solutions which, sometimes, ensures the uniqueness of the solution.