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arXiv · 2605.05670

On the inhomogeneous discounted Hamilton-Jacobi equations

Liang Jin·Jun Yan·Kai Zhao
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Abstract

In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \begin{equation}\label{hjs1} \lambda(x)u+h(x,d_x u)=c \quad \tag{$\ast$} \end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $\lambda(x)$. There is a critical value $c_0\in[-\infty,\infty)$ such that \eqref{hjs1} admits a viscosity solution if $c>c_0$ and no solution if $c c_0$. In this case, we determine the basin of the stable solution and investigate the long time behavior of the solution semigroup associated to \eqref{hjs1}. In particular, we relate the lowest convergence rate to the integral of $\lambda$ over Mather measures, which leads to an asymptotic behavior of Mather measures when $c$ goes to infinity. Assume $c\geqslant c_0$ and the equation admits a solution, we classify ergodic Mather measures and locate their distribution in the phase space.

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Liang Jin, Jun Yan, Kai Zhao. 2026-05-07. On the inhomogeneous discounted Hamilton-Jacobi equations. https://arxiv.org/abs/2605.05670

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