# State-constraint static Hamilton-Jacobi equations in nested domains

## TL;DR

This paper investigates the convergence rates of solutions to state-constraint static Hamilton-Jacobi equations in nested domains, providing optimal rates and new examples in the context of increasing domain sequences.

## Contribution

It establishes convergence rates for solutions in nested domains and demonstrates their optimality, expanding understanding of Hamilton-Jacobi equations in complex geometries.

## Key findings

- Derived explicit convergence rates for solutions
- Proved optimality of the convergence rates
- Presented new examples and discussions

## Abstract

We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}_{k \in \mathbb{N}}$ in $\mathbb{R}^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k\in \mathbb{N}$. 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 $\Omega = \bigcup_{k \in \mathbb{N}} \Omega_k$. In many cases, the rates obtained are proven to be optimal. Various new examples and discussions are provided at the end of the paper.