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 Ω_λ=(1+r(λ))Ω with a continuous function r and a positive constant λ. We characterize all solutions to the ergodic problem on Ω in terms of r. In addition, we demonstrate that the additive eigenvalue λ↦c_Ωλ on a rescaled domain Ω_λ=(1+λ)Ω 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 λ↦c_Ωλ. We provide examples where higher regularity is achieved.