Vanishing impulse cost in the quasi-variational inequality for ergodic impulse control

We consider the impulse control of a reflected diffusion. It has been proved that the long run average cost for this problem solves the ergodic Quasi-Variational inequality (Q.V.I.) $\begin{equation}\left\{\begin{array}{l}\int_{\Omega}\triangledown u_{k}\triangledown (v-u_{k})dx\geq {\int_{\Omega}}(f-\lambda _{k})\cdot(v-u_{k})dx,\\\forall_{v}\in H^{1}(\Omega),\,v\leq k+Mu_{k}\,\mathrm{a.e.;}\quad u_{k}\inH^{1}(\Omega),\,u_{k}\leq k+Mu_{k}\,\mathrm{a.e.,}\quad \int_{\Omega}u_{k}dx=0,\end{array}\right.\end{equation}$

Mu(x)=infess{c₀(ξ)+u(x+ξ);ξ≥0,x+ξ∈Ω}.

We prove uniform bounds in H¹∩L^∞ on u_k and we show that, extracting a subsequence if necessary, (u_k,λ_k) converge as k→0 to a solution of (1)₀. We also study the uniqueness of (u₀,λ₀), and we prove that it is false in general although the complete sequence λ_k converges to the maximal λ₀ such that (1)₀ admits a solution.