We study discounted Hamilton Jacobi equations on networks, without putting any restriction on their geometry. Assuming the Hamiltonians are continuous and coercive, we establish a comparison principle and provide representation formulae for solutions. We follow the approach introduced by Siconolfi and Sorrentino (2018); specifically, we associate to the differential problem on the network a discrete functional equation on an abstract underlying graph. We perform some qualitative analysis and single out a distinguished subset of vertices, called lambda-Aubry set, which shares some properties of the Aubry set for eikonal equations on compact manifolds. We finally study the asymptotic behavior of solutions and lambda-Aubry sets as the discount factor lambda becomes infinitesimal.

Discounted Hamilton-Jacobi Equations on Networks and Asymptotic Analysis

POZZA, MARCO;
2021-01-01

Abstract

We study discounted Hamilton Jacobi equations on networks, without putting any restriction on their geometry. Assuming the Hamiltonians are continuous and coercive, we establish a comparison principle and provide representation formulae for solutions. We follow the approach introduced by Siconolfi and Sorrentino (2018); specifically, we associate to the differential problem on the network a discrete functional equation on an abstract underlying graph. We perform some qualitative analysis and single out a distinguished subset of vertices, called lambda-Aubry set, which shares some properties of the Aubry set for eikonal equations on compact manifolds. We finally study the asymptotic behavior of solutions and lambda-Aubry sets as the discount factor lambda becomes infinitesimal.
2021
Hamilton-Jacobi equation
embedded networks
graphs
viscosity solutions
discrete functional equation on graphs
Hopf-Lax formula
discrete weak KAM theory
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.14085/11347
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 3
social impact