The ``ergodic limit'' for a viscous Hamilton-Jacobi equation with Dirichlet conditions (Q966208)

Summary: We study the limit, when \(\lambda\) tends to 0, of the solutions \(u_\lambda\) of the Dirichlet problem \[ -\Delta u+ \lambda u+|\nabla u|^q=f(x) \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega, \] when \(1<q\leq 2\) 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 \(u_\lambda+ \|u_\lambda\|_\infty\) converges to the boundary blow-up solution \(v_0\) of the ergodic problem associated to the stochastic optimal control with state constraint.