## Analysis & PDE

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

### Global results for eikonal Hamilton–Jacobi equations on networks

#### 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.

#### Article information

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

Dates
Revised: 25 May 2017
Accepted: 10 August 2017
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513774493

Digital Object Identifier
doi:10.2140/apde.2018.11.171

Mathematical Reviews number (MathSciNet)
MR3707295

Zentralblatt MATH identifier
1377.35250

#### Citation

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. https://projecteuclid.org/euclid.apde/1513774493

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