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.