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December  2017, 37(12): 6405-6435. doi: 10.3934/dcds.2017278

Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case

Cyril Imbert 1, and  Régis Monneau 2,

1. 

CNRS & Déepartment de Mathématiques et Applications, École Normale Supérieure (Paris), 45 rue d'Ulm, 75005 Paris, France
2. 

70, rue du Javelot, 75013 Paris, France

Received  March 2017 Published  August 2017

A multi-dimensional junction is obtained by identifying the boundaries of a finite number of copies of an Euclidian half-space. The main contribution of this article is the construction of a multidimensional vertex test function G(x, y). First, such a function has to be sufficiently regular to be used as a test function in the viscosity solution theory for quasi-convex Hamilton-Jacobi equations posed on a multi-dimensional junction. Second, its gradients have to satisfy appropriate compatibility conditions in order to replace the usual quadratic penalization function |x-y|2 in the proof of strong uniqueness (comparison principle) by the celebrated doubling variable technique. This result extends a construction the authors previously achieved in the network setting. In the multi-dimensional setting, the construction is less explicit and more delicate.

Keywords: Hamilton-Jacobi equations, multi-dimensional junctions, multi-dimensional vertex test function.
Mathematics Subject Classification: 35F21, 49L25, 35B51.
Citation: Cyril Imbert, Régis Monneau. Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6405-6435. doi: 10.3934/dcds.2017278
References:
[1]

Y. Achdou, F. Camilli, A. Cutrì and N. Tchou, Hamilton-Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445.  doi: 10.1007/s00030-012-0158-1.  Google Scholar

[2]

Y. Achdou, S. Oudet and N. Tchou, Hamilton-Jacobi equations for optimal control on junctions and networks, ESAIM Control Optim. Calc. Var., 21 (2015), 876-899.  doi: 10.1051/cocv/2014054.  Google Scholar

[3]

Y. Achdou, S. Oudet and N. Tchou, Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1091-1121, URL https://hal.archives-ouvertes.fr/hal-01162438. doi: 10.1016/j.matpur.2016.04.002.  Google Scholar

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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 (2013), 710-739.  doi: 10.1051/cocv/2012030.  Google Scholar

[5]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbb{R}^N $, SIAM J. Control Optim., 52 (2014), 1712-1744.  doi: 10.1137/130922288.  Google Scholar

[6]

G. Barles, A. Briani, E. Chasseigne and C. Imbert, Flux-limited and classical viscosity solutions for regional control problems, 2016, URL https://hal.archives-ouvertes.fr/hal-01392414, Preprint HAL 01392414. Google Scholar

[7]

G. Barles and E. Chasseigne, (Almost) everything you always wanted to know about deterministic control problems in stratified domains, Netw. Heterog. Media, 10 (2015), 809-836.  doi: 10.3934/nhm.2015.10.809.  Google Scholar

[8]

A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heterog. Media, 2 (2007), 313-331.  doi: 10.3934/nhm.2007.2.313.  Google Scholar

[9]

F. Camilli, D. Schieborn and C. Marchi, Eikonal equations on ramified spaces, Interfaces Free Bound., 15 (2013), 121-140.  doi: 10.4171/IFB/297.  Google Scholar

[10]

M. I. Freidlin and A. D. Wentzell, Diffusion processes on an open book and the averaging principle, Stochastic Process. Appl., 113 (2004), 101-126.  doi: 10.1016/j.spa.2004.03.009.  Google Scholar

[11]

Y. Giga and N. Hamamuki, Hamilton-Jacobi equations with discontinuous source terms, Comm. Partial Differential Equations, 38 (2013), 199-243.  doi: 10.1080/03605302.2012.739671.  Google Scholar

[12]

C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state-constraints sets, J. Differential Equations, 258 (2015), 1430-1460.  doi: 10.1016/j.jde.2014.11.001.  Google Scholar

[13]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 357-448.  doi: 10.24033/asens.2323.  Google Scholar

[14]

C. Imbert and V. D. Nguyen, Effective junction conditions for degenerate parabolic equations, 2016, URL https://hal.archives-ouvertes.fr/hal-01252891, Preprint HAL 01252891 (Version 2), 26 pages. Google Scholar

[15]

H. Ishii, Perron's method for Hamilton-Jacobi equations, Duke Math. J., 55 (1987), 369-384.  doi: 10.1215/S0012-7094-87-05521-9.  Google Scholar

[16]

S. Oudet, Hamilton-Jacobi equations for optimal control on multidimensional junctions, 2014, Preprint, arXiv: 1412.2679v2. Google Scholar

[17]

Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, 257 (2014), 3978-4014.  doi: 10.1016/j.jde.2014.07.015.  Google Scholar

[18]

Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman equations on multi-domains, Control and Optimization with PDE Constraints, International Series of Numerical Mathematics, 164 (2010), 93-116.  doi: 10.1007/978-3-0348-0631-2_6.  Google Scholar

show all references

References:
[1]

Y. Achdou, F. Camilli, A. Cutrì and N. Tchou, Hamilton-Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445.  doi: 10.1007/s00030-012-0158-1.  Google Scholar

[2]

Y. Achdou, S. Oudet and N. Tchou, Hamilton-Jacobi equations for optimal control on junctions and networks, ESAIM Control Optim. Calc. Var., 21 (2015), 876-899.  doi: 10.1051/cocv/2014054.  Google Scholar

[3]

Y. Achdou, S. Oudet and N. Tchou, Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1091-1121, URL https://hal.archives-ouvertes.fr/hal-01162438. doi: 10.1016/j.matpur.2016.04.002.  Google Scholar

[4]

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 (2013), 710-739.  doi: 10.1051/cocv/2012030.  Google Scholar

[5]

G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in $\mathbb{R}^N $, SIAM J. Control Optim., 52 (2014), 1712-1744.  doi: 10.1137/130922288.  Google Scholar

[6]

G. Barles, A. Briani, E. Chasseigne and C. Imbert, Flux-limited and classical viscosity solutions for regional control problems, 2016, URL https://hal.archives-ouvertes.fr/hal-01392414, Preprint HAL 01392414. Google Scholar

[7]

G. Barles and E. Chasseigne, (Almost) everything you always wanted to know about deterministic control problems in stratified domains, Netw. Heterog. Media, 10 (2015), 809-836.  doi: 10.3934/nhm.2015.10.809.  Google Scholar

[8]

A. Bressan and Y. Hong, Optimal control problems on stratified domains, Netw. Heterog. Media, 2 (2007), 313-331.  doi: 10.3934/nhm.2007.2.313.  Google Scholar

[9]

F. Camilli, D. Schieborn and C. Marchi, Eikonal equations on ramified spaces, Interfaces Free Bound., 15 (2013), 121-140.  doi: 10.4171/IFB/297.  Google Scholar

[10]

M. I. Freidlin and A. D. Wentzell, Diffusion processes on an open book and the averaging principle, Stochastic Process. Appl., 113 (2004), 101-126.  doi: 10.1016/j.spa.2004.03.009.  Google Scholar

[11]

Y. Giga and N. Hamamuki, Hamilton-Jacobi equations with discontinuous source terms, Comm. Partial Differential Equations, 38 (2013), 199-243.  doi: 10.1080/03605302.2012.739671.  Google Scholar

[12]

C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state-constraints sets, J. Differential Equations, 258 (2015), 1430-1460.  doi: 10.1016/j.jde.2014.11.001.  Google Scholar

[13]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 357-448.  doi: 10.24033/asens.2323.  Google Scholar

[14]

C. Imbert and V. D. Nguyen, Effective junction conditions for degenerate parabolic equations, 2016, URL https://hal.archives-ouvertes.fr/hal-01252891, Preprint HAL 01252891 (Version 2), 26 pages. Google Scholar

[15]

H. Ishii, Perron's method for Hamilton-Jacobi equations, Duke Math. J., 55 (1987), 369-384.  doi: 10.1215/S0012-7094-87-05521-9.  Google Scholar

[16]

S. Oudet, Hamilton-Jacobi equations for optimal control on multidimensional junctions, 2014, Preprint, arXiv: 1412.2679v2. Google Scholar

[17]

Z. Rao, A. Siconolfi and H. Zidani, Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations, J. Differential Equations, 257 (2014), 3978-4014.  doi: 10.1016/j.jde.2014.07.015.  Google Scholar

[18]

Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman equations on multi-domains, Control and Optimization with PDE Constraints, International Series of Numerical Mathematics, 164 (2010), 93-116.  doi: 10.1007/978-3-0348-0631-2_6.  Google Scholar

Figure 1.  A Hamilton-Jacobi equation posed on a multi-dimensional junction. Here there are 3 branches (or sheets — $N=3$ ) and the tangential dimension is $1$ ( $d=1$ ). We did not illustrate the junction condition on the junction hyperplane $\Gamma$ (which is a line in this example)
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$\pi_i^\pm$ of $H_i$ are also shown">Figure 2.  Monotone parts $H_i^\pm$ of a Hamiltonian $H_i$ ($H_i^-$ on the left, $H_i^+$ on the right). The Hamiltonian is in black, monotone parts in red. The tangent variable $p'$ is not shown. In this example, the minimum $A_i$ of $H_i$ is lower than $A_0$. The "inverse" functions $\pi_i^\pm$ of $H_i$ are also shown
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