Vanishing discount problem and the additive eigenvalues on changing domains

Abstract

We study the asymptotic behavior, as $\lambda \to 0^{+}$, of the state-constraint Hamilton--Jacobi equation $\varphi (\lambda )u_{\lambda }(x)+H(x,D{u}_{\lambda }(x))=0$ in $(1+r(\lambda ))\Omega$ and the corresponding additive eigenvalues, or ergodic constant $H(x,Dv(x))=c(\lambda )$ in $(1+r(\lambda ))\Omega$ with state-constraint. Here, $\Omega$ is a bounded domain of ${\mathbb{R}}^n$, $\varphi (\lambda ),r(\lambda ):(0,\infty )\to \mathbb{R}$ are continuous functions such that $\varphi$ is nonnegative and ${\lim }_{\lambda \to 0^{+}}\varphi (\lambda )={\lim }_{\lambda \to 0^{+}}r(\lambda )=0$. We obtain both convergence and non-convergence results in the convex setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue $c(\lambda )$ as $\lambda \to 0^{+}$. The main tool we use is a duality representation of solution with viscosity Mather measures.

34 pages, 3 figures. AMSart style, revision version

References in corpus (1)

• Convergence of the solutions of the discounted equation: the discrete case

Cited by in corpus (4)

• Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations
• Limit of solutions for semilinear Hamilton-Jacobi equations with degenerate viscosity
• On the regularity of stochastic effective Hamiltonian
• Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains