Global results for eikonal Hamilton–Jacobi equations on networks

Siconolfi, Antonio; Sorrentino, Alfonso

doi:10.2140/apde.2018.11.171

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Global results for eikonal Hamilton–Jacobi equations on networks

Antonio Siconolfi and Alfonso Sorrentino

Vol. 11 (2018), No. 1, 171–211

DOI: 10.2140/apde.2018.11.171

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

Vol. 11, No. 1, 2018