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Non-existence of global characteristics for viscosity solutions. (arXiv:1807.01038v1 [math.OC])
来源于:arXiv
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. In this paper we prove that there exists no other class of
integrable Hamiltonians sharing this property. To do so, we build for any
non-convex non-concave 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. 查看全文>>