Singular Dynamics for Semiconcave Functions

Abstract

Semiconcave functions are a well-known class of nonsmooth functions that possess deep connections with optimization theory and nonlinear pde’s. Their singular sets exhibit interesting structures that we investigate in this paper. First, by an energy method, we analyze the curves along which the singularities of semiconcave solutions to Hamilton–Jacobi equations propagate—the socalled generalized characteristics. This part of the paper improves the main result in [P. Albano, P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal. 162 (2002), 1–23] and simplifies the construction therein. As applications, we recover some known results for gradient flows and conservation laws. Then we derive a simple dynamics for the propagation of singularities of general semiconcave functions. This part of the work is also used to study the singularities of generalized solutions to Monge–Ampère equations. We conclude with a global propagation result for the singularities of solutions e in weak KAM theory.