The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space

Abstract

We study the asymptotic behavior of the solutions to a family of discounted Hamilton-Jacobi equations, posed in when the discount factor goes to zero. The ambient space being noncompact, we introduce an assumption implying that the Aubry set is compact and there is no degeneracy at infinity. Our approach is to deal not with a single Hamiltonian and Lagrangian but with the whole space of generalized Lagrangians, and then to define via duality minimizing measures associated with both the corresponding ergodic and discounted equations. The asymptotic result follows from the convergence properties of these measures concerning the narrow topology. We use as duality tool a separation theorem in locally convex Hausdorff spaces, and we use the strict topology in the space of the bounded generalized Lagrangians as well.