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

Let $c(\lambda)$ be the additive eigenvalue with respect to $(1+r(\lambda))\Omega$, we show that $\lambda\mapsto c(\lambda)$ is differentiable except at most a countable set, while one-sided derivatives exist everywhere. The convergence of the vanishing discount problem on changing domains is also considered, and furthermore, the limiting solution can be parametrized by a real function. Finally, we connect the regularity of this real function to the regularity of $\lambda \mapsto c(\lambda)$. Some examples are given where higher regularity is achieved.