Pathological viscosity solutions of Hamilton--Jacobi equations

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.

The authors used ChatGPT 6 Pro to assist in developing the construction of the pathological solutions through iterative discussions guided by the authors' ideas and suggestions. The authors verified all mathematical arguments and prepared the manuscript. They take full responsibility for the correctness and content of the paper