Vanishing discount selection for constant-sign local perturbations of periodic Hamiltonians
Abstract
We study the vanishing discount selection problem for critical Hamil\-ton--Jacobi equations posed on ℝd with Hamiltonian G=H−V, where H is continuous, periodic in space, convex and superlinear in the momentum, and V is a compactly supported potential of constant sign. For Vleqslant0, we prove local uniform convergence of the discounted solutions to a distinguished critical solution, which we characterize in terms of Mather measures and of the solution selected by the unperturbed periodic problem. The proof relies on a decomposition of limiting discounted measures into a retained component and a periodic component accounting for the mass lost at infinity. For Vgeqslant0, we establish convergence under the additional assumption that the unperturbed selected solution has zero average with respect to every periodic Mather measure. We also give explicit examples in which the discounted solutions converge, but natural formulas for the candidate limit fail to identify the selected solution.
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BibTeX
@misc{davini2026vanishing,
title = {{Vanishing discount selection for constant-sign local perturbations of periodic Hamiltonians}},
author = {Andrea Davini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04149},
eprint = {2610.04149},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04149},
}
RIS
TY - UNPB
TI - Vanishing discount selection for constant-sign local perturbations of periodic Hamiltonians
AU - Andrea Davini
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.04149
UR - https://arxiv.org/abs/2610.04149
AB - We study the vanishing discount selection problem for critical Hamil\-ton--Jacobi equations posed on ℝ^d with Hamiltonian G=H−V, where H is continuous, periodic in space, convex and superlinear in the momentum, and V is a compactly supported potential of constant sign. For Vleqslant0, we prove local uniform convergence of the discounted solutions to a distinguished critical solution, which we characterize in terms of Mather measures and of the solution selected by the unperturbed periodic problem. The proof relies on a decomposition of limiting discounted measures into a retained component and a periodic component accounting for the mass lost at infinity. For Vgeqslant0, we establish convergence under the additional assumption that the unperturbed selected solution has zero average with respect to every periodic Mather measure. We also give explicit examples in which the discounted solutions converge, but natural formulas for the candidate limit fail to identify the selected solution.
ER -