On the negative limit of viscosity solutions for discounted Hamilton-Jacobi equations

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