Generalized convergence of solutions for nonlinear Hamilton-Jacobi equations with state-constraint

Abstract

For a continuous Hamiltonian $H:(x,p,u)\in T^{\ast }{\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}$, we consider the asymptotic behavior of associated Hamilton--Jacobi equations with state-constraint $H(x,Du,\hat{I}»u)\le C_{\hat{I}}»$ in $\hat{I}{\text{c◯}}_{\hat{I}}»\subset {\mathbb{R}}^n$ and $H(x,Du,\hat{I}»u)\ge C_{\hat{I}}»$ on $\overline{\hat{I}}{\text{c◯}}_{\hat{I}}»\subset {\mathbb{R}}^n$ a $\hat{I}»\to 0^{+}$. When $H$ satisfies certain convex, coercive, and monotone conditions, the domain $\hat{I}{\text{c◯}}_{\hat{I}}»:=(1+r(\hat{I}»))\hat{I}\text{c◯}$ keeps bounded, star-shaped for all $\hat{I}»>0$ with ${\lim }_{\hat{I}»\to 0^{+}}r(\hat{I}»)=0$, and ${\lim }_{\hat{I}»\to 0^{+}}{C}_{\hat{I}}»=c(H)$ equals the ergodic constant of $H(\cdot ,\cdot ,0)$, we prove the convergence of solutions $u_{\hat{I}}»$ to a specific solution of the critical equation $H(x,Du,0)\le c(H)$ in $\hat{I}\text{c◯}$ and $H(x,Du,0)\ge c(H)$ on $\overline{\hat{I}}\text{c◯}$. We also discuss the generalization of such a convergence for equations with more general $C_{\hat{I}}»$ and $\hat{I}{\text{c◯}}_{\hat{I}}»$.

27 pages, some more typos corrected