Abstract We study the asymptotic behavior of solutions to the fully nonlinear Hamilton-Jacobi equation H (x, Du, u) = 0 H (x, D u, λ u) = 0 in R^n R n as 0^+ λ → 0 +. Assuming that the Aubry set is localized, we use a variational approach to derive limiting Mather-type measures and formulate a selection principle. Central to our analysis is a modified variational formula that bridges global and local state-constraint solutions, thereby extending localization techniques to the nonlinear framework.
Tu et al. (Mon,) studied this question.