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 R N , 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 to both the corresponding ergodic and discounted equations. The asymptotic result follows from convergence properties of these measures with respect to the narrow topology. We use as duality tool a separation theorem in locally convex Hausdorff spaces, we use the strict topology in the space of the bounded generalized Lagrangians as well.

Vanishing discount problem and the additive eigenvalues on changing domains

“…There are two natural normalization for solutions of (S λ ). The first one is similar to what has been considered in [11,12,19] as…”

“…The problems in bounded domains with boundary conditions were proved in [1,11]. The problem in R n under additional assumptions that lead to the compactness of the Aubry set was studied in [12]. For the selection problems with state-constraint boundary conditions, so far, there is only [11] that deals with a fixed domain, and there is not yet any result studying the changing domains situation.…”

“…We show the convergence for some natural normalization of solutions to (S λ ) together with characterizing their limits, related characterizations are done in [12] for the case the domain is R n and in [7,10,19] for the case the domain is torus T n = R n /Z n . We also discuss other related results concerning the asymptotic behavior of the additive eigenvalue of H in Ω λ and Ω λ as λ → 0 + .…”

The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 2: Nonlinear coupling

“…This together with (25) shows that w k (z) ≤ v k (z), which ensures that (27) holds. Noting that (27) combined with (26) shows that w ≤ v for all v, w ∈ V, that is, V is a singleton. The proof is complete.…”

“…Recently, there has been a great interest in the vanishing discount problem concerned with Hamilton-Jacobi equations and, furthermore, fully nonlinear degenerate elliptic PDEs. We refer to [1,5,10,13,20,[24][25][26]32] for relevant work. The asymptotic analysis in these papers relies heavily on Mather measures or their generalizations and, thus, it is considered part of Aubry-Mather and weak KAM theories.…”