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Convexity preservation for first-order nonlinear evolution equations with Neumann-type boundary conditions
Abstract
We study preservation of spatial convexity for viscosity solutions of first order Hamilton-Jacobi evolution equations with Neumann-type boundary conditions in bounded convex domains. Under a monotonicity assumption on the Hamiltonian in the outward normal direction, every viscosity supersolution of the Neumann problem is also a state-constraint supersolution. This reduces the boundary difficulty to a state-constraint problem and enables us to prove convexity preservation via the convex-envelope argument established by Alvarez, Lasry and Lions (1997). We also obtain propagation estimates for semiconvexity and strong convexity under similar structural assumptions.
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BibTeX
@misc{lin2026convexity,
title = {{Convexity preservation for first-order nonlinear evolution equations with Neumann-type boundary conditions}},
author = {Daowen Lin and Qing Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11509},
eprint = {2610.11509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11509},
}
RIS
TY - UNPB
TI - Convexity preservation for first-order nonlinear evolution equations with Neumann-type boundary conditions
AU - Daowen Lin
AU - Qing Liu
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.11509
UR - https://arxiv.org/abs/2610.11509
AB - We study preservation of spatial convexity for viscosity solutions of first order Hamilton-Jacobi evolution equations with Neumann-type boundary conditions in bounded convex domains. Under a monotonicity assumption on the Hamiltonian in the outward normal direction, every viscosity supersolution of the Neumann problem is also a state-constraint supersolution. This reduces the boundary difficulty to a state-constraint problem and enables us to prove convexity preservation via the convex-envelope argument established by Alvarez, Lasry and Lions (1997). We also obtain propagation estimates for semiconvexity and strong convexity under similar structural assumptions.
ER -