Lower-dimensional autonomous examples

Determine whether the pathological viscosity-solution phenomenon established for smooth autonomous Hamiltonians in dimension four also occurs in dimension two or dimension three; specifically, establish whether there exist a continuous nowhere-differentiable function and a bounded smooth autonomous Hamiltonian on a two- or three-dimensional domain such that the function solves the Hamilton–Jacobi equation with multiple constant right-hand sides in the viscosity sense.

Background

The paper constructs a continuous nowhere-differentiable function in four dimensions and a smooth autonomous Hamiltonian for which the same function is a viscosity solution of H(Du)=c for every c in [1,2]. The construction lifts a specially designed two-dimensional function into four dimensions by adding two spatial variables.

The authors show that the analogous nonautonomous phenomenon already occurs in dimension two and explain that dimension one cannot support such an example under the stated assumptions. They therefore identify dimensions two and three as the remaining unresolved cases for the autonomous result, noting that their lifting construction cannot obtain a two-dimensional autonomous example from a one-dimensional function.