State-Constraint Static Hamilton-Jacobi Equations in Nested Domains

Yeoneung Kim, Hung V. Tran, Son N. Tu

Abstract: 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 $k\in \Bbb{N}$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega = \bigcup_{k \in \Bbb{N}} \Omega_k$. In many cases, the rates obtained are proven to be optimal. Various new examples and discussions are provided at the end of the paper.