Analysis & PDE

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

Global results for eikonal Hamilton–Jacobi equations on networks

Antonio Siconolfi and Alfonso Sorrentino

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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
Received: 5 December 2016
Revised: 25 May 2017
Accepted: 10 August 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 35F21: Hamilton-Jacobi equations 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)
Secondary: 35B51: Comparison principles 49L25: Viscosity solutions

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

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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References

  • Y. Achdou, F. Camilli, A. Cutr\`\i, and N. Tchou, “Hamilton–Jacobi equations constrained on networks”, NoDEA Nonlinear Differential Equations Appl. 20:3 (2013), 413–445.
  • M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations, Birkhäuser, Boston, 1997.
  • G. Barles, Solutions de viscosité des équations de Hamilton–Jacobi, Mathématiques & Applications 17, Springer, 1994.
  • G. Barles, A. Briani, and E. Chasseigne, “A Bellman approach for two-domains optimal control problems in $\mathbb R^N$”, ESAIM Control Optim. Calc. Var. 19:3 (2013), 710–739.
  • G. Barles, A. Briani, and E. Chasseigne, “A Bellman approach for regional optimal control problems in $\mathbb{R}^N$”, SIAM J. Control Optim. 52:3 (2014), 1712–1744.
  • P. Bernard and B. Buffoni, “The Monge problem for supercritical Mañé potentials on compact manifolds”, Adv. Math. 207:2 (2006), 691–706.
  • P. Bernard and B. Buffoni, “Optimal mass transportation and Mather theory”, J. Eur. Math. Soc. $($JEMS$)$ 9:1 (2007), 85–121.
  • A. Bressan and Y. Hong, “Optimal control problems on stratified domains”, Netw. Heterog. Media 2:2 (2007), 313–331.
  • F. Camilli and C. Marchi, “A comparison among various notions of viscosity solution for Hamilton–Jacobi equations on networks”, J. Math. Anal. Appl. 407:1 (2013), 112–118.
  • F. Camilli, D. Schieborn, and C. Marchi, “Eikonal equations on ramified spaces”, Interfaces Free Bound. 15:1 (2013), 121–140.
  • A. Davini, A. Fathi, R. Iturriaga, and M. Zavidovique, “Convergence of the solutions of the discounted equation: the discrete case”, Math. Z. 284:3-4 (2016), 1021–1034.
  • A. Fathi, “Weak KAM Theorem in Lagrangian dynamics”, lecture notes, 2008, https://www.math.u-bordeaux.fr/~pthieull/Recherche/KamFaible/Publications/Fathi2008_01.pdf.
  • A. Fathi and A. Siconolfi, “Existence of $C^1$ critical subsolutions of the Hamilton–Jacobi equation”, Invent. Math. 155:2 (2004), 363–388.
  • A. Fathi and A. Siconolfi, “PDE aspects of Aubry–Mather theory for quasiconvex Hamiltonians”, Calc. Var. Partial Differential Equations 22:2 (2005), 185–228.
  • G. Galise, C. Imbert, and R. Monneau, “A junction condition by specified homogenization and application to traffic lights”, Anal. PDE 8:8 (2015), 1891–1929.
  • M. Garavello and B. Piccoli, Traffic flow on networks: conservation laws models, AIMS Series on Applied Mathematics 1, American Institute of Mathematical Sciences, Springfield, MO, 2006.
  • D. A. Gomes, “Viscosity solution methods and the discrete Aubry–Mather problem”, Discrete Contin. Dyn. Syst. 13:1 (2005), 103–116.
  • C. Imbert and R. Monneau, “Quasi-convex Hamilton–Jacobi equations posed on junctions: the multi-dimensional case”, preprint, 2016.
  • C. Imbert and R. Monneau, “Flux-limited solutions for quasi-convex Hamilton–Jacobi equations on networks”, Ann. Sci. Éc. Norm. Supér. $(4)$ 50:2 (2017), 357–448.
  • C. Imbert, R. Monneau, and H. Zidani, “A Hamilton–Jacobi approach to junction problems and application to traffic flows”, ESAIM Control Optim. Calc. Var. 19:1 (2013), 129–166.
  • H. Ishii, “A simple, direct proof of uniqueness for solutions of the Hamilton–Jacobi equations of eikonal type”, Proc. Amer. Math. Soc. 100:2 (1987), 247–251.
  • P.-L. Lions and P. Souganidis, “Viscosity solutions for junctions: well posedness and stability”, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27:4 (2016), 535–545.
  • J. Nash, “The imbedding problem for Riemannian manifolds”, Ann. of Math. $(2)$ 63 (1956), 20–63.
  • Y. V. Pokornyi and A. V. Borovskikh, “Differential equations on networks (geometric graphs)”, J. Math. Sci. $($N. Y.$)$ 119:6 (2004), 691–718.
  • Z. Rao, A. Siconolfi, and H. Zidani, “Transmission conditions on interfaces for Hamilton–Jacobi–Bellman equations”, J. Differential Equations 257:11 (2014), 3978–4014.
  • D. Schieborn and F. Camilli, “Viscosity solutions of Eikonal equations on topological networks”, Calc. Var. Partial Differential Equations 46:3-4 (2013), 671–686.
  • H. M. Soner, “Optimal control with state-space constraint, I”, SIAM J. Control Optim. 24:3 (1986), 552–561.
  • A. Sorrentino, Action-minimizing methods in Hamiltonian dynamics: an introduction to Aubry–Mather theory, Mathematical Notes 50, Princeton University Press, 2015.
  • X. Su and P. Thieullen, “Convergence of discrete Aubry–Mather model in the continuous limit”, preprint, 2015.
  • T. Sunada, Topological crystallography: with a view towards discrete geometric analysis, Surveys and Tutorials in the Applied Mathematical Sciences 6, Springer, 2013.
  • M. Zavidovique, “Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory”, J. Mod. Dyn. 4:4 (2010), 693–714.
  • M. Zavidovique, “Strict sub-solutions and Mañé potential in discrete weak KAM theory”, Comment. Math. Helv. 87:1 (2012), 1–39.