Global propagation of singularities for discounted Hamilton-Jacobi equations

The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation

\begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document}

with fixed constant \begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}. We reduce the problem for equation \begin{document}$(\mathrm{HJ}_{\lambda})$\end{document} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of \begin{document}$(\mathrm{HJ}_{\lambda})$\end{document} propagate along locally Lipschitz singular characteristics \begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document} and time \begin{document}$ t $\end{document} can extend to \begin{document}$ +\infty $\end{document}. Essentially, we use \begin{document}$ \sigma $\end{document}-compactness of the Euclidean space which is different from the original construction in [4]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of \begin{document}$ u $\end{document} and the complement of Aubry set using the basic idea from [9].