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

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 ${\mathrm{\Omega }}_{\lambda }=(1+r(\lambda ))\mathrm{\Omega }$ with a continuous function $r$ and a positive constant $\lambda$. We characterize all solutions to the ergodic problem on $\mathrm{\Omega }$ in terms of $r$. In addition, we demonstrate that the additive eigenvalue $\lambda \mapsto c_{{\mathrm{\Omega }}_{\lambda }}$ on a rescaled domain ${\mathrm{\Omega }}_{\lambda }=(1+\lambda )\mathrm{\Omega }$ 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 $\lambda \mapsto c_{{\mathrm{\Omega }}_{\lambda }}$. We provide examples where higher regularity is achieved.