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2020 Nonexistence of global characteristics for viscosity solutions
Valentine Roos
Anal. PDE 13(4): 1145-1172 (2020). DOI: 10.2140/apde.2020.13.1145

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.

Citation

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Valentine Roos. "Nonexistence of global characteristics for viscosity solutions." Anal. PDE 13 (4) 1145 - 1172, 2020. https://doi.org/10.2140/apde.2020.13.1145

Information

Received: 22 August 2018; Revised: 23 March 2019; Accepted: 18 April 2019; Published: 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07221200
MathSciNet: MR4109903
Digital Object Identifier: 10.2140/apde.2020.13.1145

Subjects:
Primary: 49L25

Rights: Copyright © 2020 Mathematical Sciences Publishers

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