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

Abstract

We study the limit, when λ tends to 0, of the solutions u\_λ of the Dirichlet problem \-∆\_u + λu + |∇\_u\_|q = f(x) in Ω u = 0 on ∂ Ω, when 1 < q ≤ 2 and f is bounded. In case the limit problem does not have any solution, we prove that \_u\_λ has a complete blow-up (\_u\_λ → -∞) and its behaviour is described in terms of the corresponding ergodic problem with state constraint conditions. In particular, \_λu\_λ converges to the ergodic constant \_c\_0 and \_u\_λ + ||\_u\_λ||∞ converges to the boundary blow-up solution \_ν\_0 of the ergodic problem associated to the stochastic optimal control with state constraint.