Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 1 (2018), 171-211.

Global results for eikonal Hamilton–Jacobi equations on networks

Antonio Siconolfi and Alfonso Sorrentino

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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.

Article information

Anal. PDE, Volume 11, Number 1 (2018), 171-211.

Received: 5 December 2016
Revised: 25 May 2017
Accepted: 10 August 2017
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35F21: Hamilton-Jacobi equations 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)
Secondary: 35B51: Comparison principles 49L25: Viscosity solutions

Hamilton–Jacobi equation embedded networks graphs viscosity solutions viscosity subsolutions comparison principle discrete functional equation on graphs Hopf–Lax formula discrete weak KAM theory


Siconolfi, Antonio; Sorrentino, Alfonso. Global results for eikonal Hamilton–Jacobi equations on networks. Anal. PDE 11 (2018), no. 1, 171--211. doi:10.2140/apde.2018.11.171.

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