Pathological viscosity solutions of Hamilton--Jacobi equations

Abstract: We construct a continuous function $u$ on the closed unit ball in $\mathbb{R}<sup>4$ and a smooth autonomous Hamiltonian $H:\mathbb{R}<sup>4\to[1,2]$ 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.