Generalized convergence of solutions for nonlinear Hamilton-Jacobi equations with state-constraint

For a continuous Hamiltonian $H:(x,p,u)\in T^{\ast }{\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}$, we consider the asymptotic behavior of associated Hamilton–Jacobi equations with state-constraint $H(x,Du,\lambda u)\le C_{\lambda }$ in ${\mathrm{\Omega }}_{\lambda }\subset {\mathbb{R}}^n$ and $H(x,Du,\lambda u)\ge C_{\lambda }$ on ${\bar{\mathrm{\Omega }}}_{\lambda }\subset {\mathbb{R}}^n$ a $\lambda \to 0^{+}$. When $H$ satisfies certain convex, coercive, and monotone conditions, the domain ${\mathrm{\Omega }}_{\lambda }:=(1+r(\lambda ))\mathrm{\Omega }$ keeps bounded, star-shaped for all $\lambda >0$ with ${\lim }_{\lambda \to 0^{+}}r(\lambda )=0$, and ${\lim }_{\lambda \to 0^{+}}{C}_{\lambda }=c(H)$ equals the ergodic constant of $H(\cdot ,\cdot ,0)$, we prove the convergence of solutions $u_{\lambda }$ to a specific solution of the critical equation $H(x,Du,0)\le c(H)$ in $\mathrm{\Omega }$ and $H(x,Du,0)\ge c(H)$ on $\bar{\mathrm{\Omega }}$. We also discuss the generalization of such a convergence for equations with more general $C_{\lambda }$ and ${\mathrm{\Omega }}_{\lambda }$.