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 $H(x, Du, λu) \leq C_λ$ in $Ω_λ\subset \mathbb{R}^n$ and $H(x, Du, λu) \geq C_λ$ on $\overlineΩ_λ\subset \mathbb{R}^n$ a $λ\rightarrow 0^+$. When $H$ satisfies certain convex, coercive, and monotone conditions, the domain $Ω_λ:=(1+r(λ))Ω$ keeps bounded, star-shaped for all $λ>0$ with $\lim_{λ\rightarrow 0^+}r(λ)=0$, and $\lim_{λ\rightarrow 0^+}C_λ=c(H)$ equals the ergodic constant of $H(\cdot,\cdot,0)$, we prove the convergence of solutions $u_λ$ to a specific solution of the critical equation $H(x, Du, 0)\leq c(H) $ in $Ω$ and $H(x, Du, 0)\geq c(H) $ on $\overlineΩ$. We also discuss the generalization of such a convergence for equations with more general $C_λ$ and $Ω_λ$.
27 pages, some more typos corrected