Global propagation of singularities for discounted Hamilton-Jacobi equations

TL;DR

This paper investigates how singularities in viscosity solutions of discounted Hamilton-Jacobi equations propagate globally, establishing their extension along Lipschitz characteristics and relating their structure to the Aubry set.

Contribution

It reduces the problem to a time-dependent case and proves the global propagation of singularities along Lipschitz characteristics, linking the singular set to the Aubry set.

Findings

Singularities propagate along locally Lipschitz characteristics extending to infinity.

Established homotopy equivalence between the singular set and the complement of the Aubry set.

Reduced the problem to a time-dependent Hamilton-Jacobi equation for analysis.

Abstract

The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation \begin{equation}\label{eq:discount 1}\tag{HJ$_\lambda$} \lambda v(x)+H( x, Dv(x) )=0 , \quad x\in \mathbb{R}^n. \end{equation} We reduce the problem for equation \eqref{eq:discount 1} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We proved that the singularities of the viscosity solution of \eqref{eq:discount 1} propagate along locally Lipschitz singular characteristics which can extend to $+\infty$. We also obtained the homotopy equivalence between the singular set and the complement of associated the Aubry set with respect to the viscosity solution of equation \eqref{eq:discount 1}.