State-constraint static Hamilton-Jacobi equations in nested domains

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.

25 pages, 1 figure, typos corrected, final version

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