Two different types of generalized solutions, namely viscosity and variational
solutions, were introduced to solve the first-order evolutionary Hamilton–Jacobi
equation. They coincide if the Hamiltonian is convex in the momentum variable. We
prove that there exists no other class of integrable Hamiltonians sharing this
property. To do so, we build for any nonconvex, nonconcave integrable Hamiltonian
a smooth initial condition such that the graph of the viscosity solution is
not contained in the wavefront associated with the Cauchy problem. The
construction is based on a new example for a saddle Hamiltonian and a precise
analysis of the one-dimensional case, coupled with reduction and approximation
arguments.
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