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

For a continuous Hamiltonian $H : (x, p, u) \in T^*\mathbb{R}^n \times \mathbb{R}\rightarrow \mathbb{R}$, we consider the asymptotic behavior of associated Hamilton--Jacobi equations with state-constraint $H(x, Du, \lambda u) \leq C_\lambda$ in $\Omega_\lambda\subset \mathbb{R}^n$ and $H(x, Du, \lambda u) \geq C_\lambda$ on $\overline{\Omega}_\lambda\subset \mathbb{R}^n$ a $\lambda\rightarrow 0^+$. When $H$ satisfies certain convex, coercive, and monotone conditions, the domain $\Omega_\lambda:=(1+r(\lambda))\Omega$ keeps bounded, star-shaped for all $\lambda>0$ with $\lim_{\lambda\rightarrow 0^+}r(\lambda)=0$, and $\lim_{\lambda\rightarrow 0^+}C_\lambda=c(H)$ 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)\leq c(H) $ in $\Omega$ and $H(x, Du, 0)\geq c(H) $ on $\overline{\Omega}$. We also discuss the generalization of such a convergence for equations with more general $C_\lambda$ and $\Omega_\lambda$.