Excess action and broken characteristics for Hamilton–Jacobi equations

Abstract

We study propagation of singularities for Hamilton–Jacobi equations St+H(t,x,∇S)=0,(t,x)∈(0,∞)×Rn, by means of the excess Lagrangian action and a related class of characteristics. In a sense, the excess action gauges how far a curve X(t) is from being action minimizing for a given viscosity solution S(t,x) of the Hamilton–Jacobi equation. Broken characteristics are defined as curves along which the excess action grows at the slowest pace possible. In particular, we demonstrate that broken characteristics carry the singularities of the viscosity solution.