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Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces
Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
S. Oudet, Hamilton-Jacobi equations for optimal control on multidimensional junctions, 2014, Preprint, arXiv: 1412.2679v2. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
S. Oudet, Hamilton-Jacobi equations for optimal control on multidimensional junctions, 2014, Preprint, arXiv: 1412.2679v2. |
[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. |
[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. |


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