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Homogenization of Hamilton-Jacobi equations on networks

Marco Pozza, Antonio Siconolfi, Alfonso Sorrentino

Abstract

We prove a homogenization result for a family of time-dependent Hamilton-Jacobi equations, rescaled by a parameter $\varepsilon$ tending to zero, posed on a periodic network, with a suitable notion of periodicity that will be defined. As $\varepsilon$ becomes infinitesimal, we derive a limiting Hamilton-Jacobi equation in a Euclidean space, whose dimension is determined by the topological complexity of the network and is independent of the ambient space in which the network is embedded. Among the key contributions of our analysis, we extend to the setting of networks and graphs Mather's result on the asymptotic behavior of the average minimal action functional, as time tends to infinity. Additionally, we establish the well-posedness of the approximating problems, representing a nontrivial generalization of existing results for finite networks to a non-compact setting.

Homogenization of Hamilton-Jacobi equations on networks

Abstract

We prove a homogenization result for a family of time-dependent Hamilton-Jacobi equations, rescaled by a parameter tending to zero, posed on a periodic network, with a suitable notion of periodicity that will be defined. As becomes infinitesimal, we derive a limiting Hamilton-Jacobi equation in a Euclidean space, whose dimension is determined by the topological complexity of the network and is independent of the ambient space in which the network is embedded. Among the key contributions of our analysis, we extend to the setting of networks and graphs Mather's result on the asymptotic behavior of the average minimal action functional, as time tends to infinity. Additionally, we establish the well-posedness of the approximating problems, representing a nontrivial generalization of existing results for finite networks to a non-compact setting.

Paper Structure

This paper contains 36 sections, 426 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main novelties and strategy
  3. Comparison with previous literature
  4. Organization of the article
  5. Periodic networks and main theorem
  6. Networks
  7. Abstract graphs and topological crystals
  8. Underlying abstract graph of a network and periodic networks
  9. Hamiltonians on networks and statement of the Main Theorem
  10. Time-dependent Hamilton--Jacobi equations on periodic networks
  11. Notions of solution, subsolution and supersolution
  12. Existence and uniqueness of solutions to the evolutive Hamilton-Jacobi equation
  13. Overall Lagrangian on N
  14. Action functional in N
  15. Lax--Oleinik representation formula and comparison result
  16. ...and 21 more sections

Figures (5)

  • Figure 1: A periodic network \ref{['subfig:2bouquetcry']} over the 2--bouquet \ref{['subfig:2bouquet']}.
  • Figure 2: The honeycomb network in \ref{['subfig:graphenecry']} is a periodic graph over the base graph \ref{['subfig:graphenebasecry']}.
  • Figure 3: We call $\Gamma_1$, $\Gamma_2$ the (oriented) graphs \ref{['subfig:graphenebase']}, \ref{['subfig:bouquet']} and ${\mathbf T}_1$, ${\mathbf T}_2$ their spanning trees in \ref{['subfig:graphenetree']}, \ref{['subfig:bouquettree']}, respectively.
  • Figure 4: In \ref{['subfig:graphene']} is represented a maximal topological graph $\widehat{\Gamma}_1$ of the graph $\Gamma_1$ in \ref{['subfig:graphenebasecov']}. In \ref{['subfig:graphenelift']} it is shown a lift of $\Gamma_1$.
  • Figure 5: In \ref{['subfig:square']} it is represented an embedding in ${\mathbb R}^2$ of a finite graph $\Gamma_0$ and in \ref{['subfig:squaretree']} the corresponding embedding of a spanning tree of $\Gamma_0$, which we call ${\mathcal{N}}_{\mathbf T}'$. Using the process described above, we get in \ref{['subfig:squarecov']} the relative embedding in ${\mathbb R}^3$ of $\Gamma$. As noted in Remark \ref{['covnetbuild']}, it is made of infinite copies of ${\mathcal{N}}_{\mathbf T}'$.

Theorems & Definitions (40)

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  • ...and 30 more