The ‘‘ergodic limit’’ for a viscous Hamilton−Jacobi equation with Dirichlet conditions

Abstract

We study the limit, when $\lambda$ tends to $0$, of the solutions $u_{\lambda }$ of the Dirichlet problem

when $1<q\le 2$ and $f$ is bounded. In case the limit problem does not have any solution, we prove that $u_{\lambda }$ has a complete blow-up $(u_{\lambda }\to -\infty )$ and its behaviour is described in terms of the corresponding ergodic problem with state constraint conditions. In particular, $\lambda u_{\lambda }$ converges to the ergodic constant $c_0$ and $u_{\lambda }+\Vert u_{\lambda }{\Vert }_{\infty }$ converges to the boundary blow-up solution ${\nu }_0$ of the ergodic problem associated to the stochastic optimal control with state constraint.