Vanishing discount problems for Hamilton--Jacobi equations on changing domains

We study the asymptotic behavior, as $\lambda\rightarrow 0^+$, of the Hamilton-Jacobi equation $\phi(\lambda) u_\lambda(x) + H(x,Du_\lambda(x)) = 0$ in $(1+r(\lambda))\Omega$ with state-constraint boundary condition. Here, $\Omega$ is a bounded domain of $ \mathbb{R}^n$, $\phi(\lambda), r(\lambda):(0,\infty)\rightarrow \mathbb{R}$ are continuous functions such that $\phi$ is nonnegative and $\lim_{\lambda\rightarrow 0^+} \phi(\lambda) = \lim_{\lambda\rightarrow 0^+} r(\lambda) = 0$. Surprisingly, we are able to obtain both convergence results and non-convergence results in this convex setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue of $H$ in $(1+r(\lambda))\Omega$ as $\lambda\rightarrow 0^+$. The main tool we use is a duality representation of solution with viscosity Mather measures.Comment: 31 pages, 3 figures. AMSart style, errors fixe