Vol. 11, No. 1, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Global results for eikonal Hamilton–Jacobi equations on networks

Antonio Siconolfi and Alfonso Sorrentino

Vol. 11 (2018), No. 1, 171–211
Abstract

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.

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
Mathematical Subject Classification 2010
Primary: 35F21, 35R02
Secondary: 35B51, 49L25
Milestones
Received: 5 December 2016
Revised: 25 May 2017
Accepted: 10 August 2017
Published: 17 September 2017
Authors
Antonio Siconolfi
Department of Mathematics
Sapienza Università di Roma
Rome
Italy
Alfonso Sorrentino
Department of Mathematics
Università degli Studi di Roma “Tor Vergata”
Rome
Italy