The regularity with respect to domains of the additive eigenvalues of superquadratic Hamilton--Jacobi equation

TL;DR

This paper investigates how additive eigenvalues of superquadratic Hamilton--Jacobi equations vary with changing domains, establishing convergence, regularity properties, and characterizing solutions in relation to domain scaling.

Contribution

It characterizes solutions to the ergodic problem on scaled domains and links the regularity of eigenvalues to the regularity of a parameterized solution.

Findings

Additive eigenvalues have one-sided derivatives everywhere on scaled domains.

The limiting solution can be parameterized by a real function.

Higher regularity of the solution is achievable under certain conditions.

Abstract

We study the additive eigenvalues on changing domains, along with the associated vanishing discount problems. We consider the convergence of the vanishing discount problem on changing domains for a general scaling type $\Omega_\lambda = (1+r(\lambda))\Omega$ with a continuous function $r$ and a positive constant $\lambda$. We characterize all solutions to the ergodic problem on $\Omega$ in terms of $r$. In addition, we demonstrate that the additive eigenvalue $\lambda\mapsto c_{\Omega_\lambda}$ on a rescaled domain $\Omega_\lambda = (1+\lambda)\Omega$ possesses one-sided derivatives everywhere. Additionally, the limiting solution can be parameterized by a real function, and we establish a connection between the regularity of this real function and the regularity of $\lambda \mapsto c_{\Omega_\lambda}$. We provide examples where higher regularity is achieved.