Pathological viscosity solutions of Hamilton--Jacobi equations

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.