State-constraint static Hamilton-Jacobi equations in nested domains

We study state-constraint static Hamilton--Jacobi equations in a sequence of domains $\{\Omega_k\}_{k \in \Bbb{N}}$ in $\Bbb{R}^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k\in \Bbb{N}$. ...