Pathological viscosity solutions of Hamilton--Jacobi equations

Hung V. Tran; Yifeng Yu

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arXiv · 2610.05736

Pathological viscosity solutions of Hamilton--Jacobi equations

Hung V. Tran·Yifeng Yu

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Abstract

We construct a continuous function $u$ on the closed unit ball in $\mathbb{R}^4$ and a smooth autonomous Hamiltonian $H:\mathbb{R}^4\to[1,2]$ for which $u$ solves $H(Du)=c$ in the unit ball in the viscosity sense for every $c\in[1,2]$. 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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Hung V. Tran, Yifeng Yu. 2026-10-05. Pathological viscosity solutions of Hamilton--Jacobi equations. https://arxiv.org/abs/2610.05736

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