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

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