2022/05/10 by Dolcetta, Italo Capuzzo, Davini, Andrea · 2 citations
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.04920
We study the asymptotic behavior of the viscosity solutions uλG of the Hamilton-Jacobi (HJ) equation λu(x)+G(x,u')=c(G) \hboxin ℝ as the positive discount factor λ tends to 0, where G(x,p):=H(x,p)-V(x) is the perturbation of a Hamiltonian H∈ C(\mathbb R×\mathbb R), \mathbb Z-periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential V∈ Cc(\mathbb R). The constant c(G) appearing above is defined as the infimum of values a∈ \mathbb R for which the HJ equation G(x,u')=a in \mathbb R admits bounded viscosity subsolutions. We prove that the functions uλG locally uniformly converge, for λ→ 0+, to a specific solution uG0 of the critical equation G(x,u')=c(G) \hboxin \mathbb R. We identify u0G in terms of projected Mather measures for G and of the limit u0H to the unperturbed periodic problem. This can be regarded as an extension to a noncompact setting of the main results in [17]. Our work also includes a qualitative analysis of \eqrefabs with a weak KAM theoretic flavor.