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