Propagation of singularities for solutions of nonlinear first order partial differential equations (Q1599486)

This paper deals with new techniques to analyse the singular set of weak solutions to the nonlinear first order PDE of the form \(F(x,u,Du)=0\). The solution \(u(x)\) is supposed to be semiconcave (i.e. \(u\) can be locally represented as a concave function plus a quadratic polynomial) and \(F(x,u,p)\) is a nonlinear convex function with respect to \(p\). The points of discontinuity of \(Du(x)\) are called singularities of \(u(x)\). The corresponding singular set of \(u\) is denoted by \(\Sigma(u)\). A singularity of \(u\) at the point \(x_0\) propagates if the connected component of \(\Sigma(u)\) containing \(x_0\) does not reduce to the singleton \(\{x_0\}\). The main objective achieved here is a classification of the singularities of \(u\) that propagate along Lipschitz arcs.