On the Negative Limit of Viscosity Solutions for Discounted Hamilton–Jacobi Equations

Suppose M is a closed Riemannian manifold. For $$C^2$$ generic (in the sense of Mañé) Tonelli Hamiltonian $$H: T^*M\rightarrow \mathbb {R}$$ , minimal viscosity solution $$u_\lambda ^-:M\rightarrow negative discounted equation $$\begin{aligned} -\lambda u+H(x,d_xu)=c(H),\quad x\in M,\ \lambda >0 \end{aligned}$$ with Mañé’s critical value c(H) converges to uniquely established $$u_0^-$$ Hamilton–Jacobi H(x,d_x u)=c(H),\quad as $$\lambda \rightarrow 0_+$$ . We also propose dynamical interpretation