Vol. 13, No. 4, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 6, 1605–1954
Issue 5, 1269–1603
Issue 4, 945–1268
Issue 3, 627–944
Issue 2, 317–625
Issue 1, 1–316

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
Nonexistence of global characteristics for viscosity solutions

Valentine Roos

Vol. 13 (2020), No. 4, 1145–1172
Abstract

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.

Keywords
Hamilton–Jacobi equation, nonconvex Hamiltonian dynamics, viscosity solution, variational solution, wavefronts, characteristics
Mathematical Subject Classification 2010
Primary: 49L25
Milestones
Received: 22 August 2018
Revised: 23 March 2019
Accepted: 18 April 2019
Published: 13 June 2020
Authors
Valentine Roos
UMPA
ENS Lyon
Lyon
France