arXiv 2106.015462021-06-11math.DSmath.AP

Global propagation of singularities for discounted Hamilton-Jacobi equations

Cui Chen, Jiahui Hong, Kai Zhao

英文摘要：

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$_λ$} λv(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}.

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