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Pathological viscosity solutions of Hamilton--Jacobi equations
Abstract
We construct a continuous function u on the closed unit ball in ℝ4 and a smooth autonomous Hamiltonian H:ℝ4→[1,2] for which u solves H(Du)=c in the unit ball in the viscosity sense for every c∈[1,2]. This reveals a striking and distinctive pathology of viscosity solutions: a fixed function and a fixed Hamiltonian need not determine the right-hand side of the equation.
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BibTeX
@misc{tran2026pathological,
title = {{Pathological viscosity solutions of Hamilton--Jacobi equations}},
author = {Hung V. Tran and Yifeng Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05736},
eprint = {2610.05736},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05736},
}
RIS
TY - UNPB
TI - Pathological viscosity solutions of Hamilton--Jacobi equations
AU - Hung V. Tran
AU - Yifeng Yu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05736
UR - https://arxiv.org/abs/2610.05736
AB - We construct a continuous function u on the closed unit ball in ℝ^4 and a smooth autonomous Hamiltonian H:ℝ^4→[1,2] for which u solves H(Du)=c in the unit ball in the viscosity sense for every c∈[1,2]. This reveals a striking and distinctive pathology of viscosity solutions: a fixed function and a fixed Hamiltonian need not determine the right-hand side of the equation.
ER -