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 Omega(lambda) =(1 + r(lambda))Omega with a continuous function rand 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 bar right arrow 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 bar right arrowc(Omega lambda). We provide examples where higher regularity is achieved. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.