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Global results for
eikonal Hamilton–Jacobi equations on networks
Antonio Siconolfi and Alfonso Sorrentino |
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Vol. 11 (2018), No. 1, 171–211
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Abstract |
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We study a one-parameter family of eikonal Hamilton–Jacobi equations on an embedded network, and prove that there exists a unique critical value for which the corresponding equation admits global solutions, in a suitable viscosity sense. Such a solution is identified, via a Hopf–Lax-type formula, once an admissible trace is assigned on an intrinsic boundary. The salient point of our method is to associate to the network an abstract graph, encoding all of the information on the complexity of the network, and to relate the differential equation to a discrete functional equation on the graph. Comparison principles and representation formulae are proven in the supercritical case as well. |
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Keywords
Hamilton–Jacobi equation, embedded networks, graphs,
viscosity solutions, viscosity subsolutions, comparison
principle, discrete functional equation on graphs, Hopf–Lax
formula, discrete weak KAM theory
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Mathematical Subject Classification 2010
Primary: 35F21, 35R02
Secondary: 35B51, 49L25
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Milestones
Received: 5 December 2016
Revised: 25 May 2017
Accepted: 10 August 2017
Published: 17 September 2017
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Authors |
| Antonio Siconolfi | |
| Department of Mathematics Sapienza Università di Roma Rome Italy |
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| Alfonso Sorrentino | |
| Department of Mathematics Università degli Studi di Roma “Tor Vergata” Rome Italy |
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