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arXiv · 2610.11509

Convexity preservation for first-order nonlinear evolution equations with Neumann-type boundary conditions

Daowen Lin·Qing Liu
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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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Daowen Lin, Qing Liu. 2026-10-08. Convexity preservation for first-order nonlinear evolution equations with Neumann-type boundary conditions. https://arxiv.org/abs/2610.11509

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