Y. Achdou, F. Camilli, A. Cutr-`-cutr-`-i, and N. Tchou, Hamilton???Jacobi equations constrained on networks, Nonlinear Differential Equations and Applications NoDEA, vol.62, issue.1, pp.413-445, 2013.
DOI : 10.1007/BF01396221
URL : https://hal.archives-ouvertes.fr/hal-00656919

B. Andreianov and K. Sbihi, Strong Boundary Traces and Well-Posedness for Scalar Conservation Laws with Dissipative Boundary Conditions, Hyperbolic problems: theory, numerics, applications, pp.937-945, 2008.
DOI : 10.1007/978-3-540-75712-2_98
URL : https://hal.archives-ouvertes.fr/hal-00475870

B. Andreianov and K. Sbihi, Scalar conservation laws with nonlinear boundary conditions, Comptes Rendus Mathematique, vol.345, issue.8, pp.431-434, 2007.
DOI : 10.1016/j.crma.2007.09.008
URL : https://hal.archives-ouvertes.fr/hal-00475852

B. Andreianov and K. Sbihi, Well-posedness of general boundary-value problems for scalar conservation laws, Transactions of the American Mathematical Society, vol.367, issue.6, pp.3763-3806, 2015.
DOI : 10.1090/S0002-9947-2015-05988-1
URL : https://hal.archives-ouvertes.fr/hal-00708973

G. Barles, A. Briani, and E. Chasseigne, A Bellman Approach for Regional Optimal Control Problems in $\mathbb{R}^N$, SIAM Journal on Control and Optimization, vol.52, issue.3, pp.1712-1744, 2014.
DOI : 10.1137/130922288

G. Barles and A. Briani, Emmanuel Chasseigne, and Cyril Imbert. Flux-limited and classical viscosity solutions for regional control problems, 2016.

M. Boué, P. Dupuis, and R. S. Ellis, Large deviations for small noise diffusions with discontinuous statistics. Probab. Theory Related Fields, pp.125-149, 2000.

F. Camilli, C. Marchi, and D. Schieborn, The vanishing viscosity limit for Hamilton???Jacobi equations on networks, Journal of Differential Equations, vol.254, issue.10, pp.4122-4143, 2013.
DOI : 10.1016/j.jde.2013.02.013

F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, vol.178, 1998.
URL : https://hal.archives-ouvertes.fr/hal-00863298

M. G. Crandall, L. C. Evans, and P. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, vol.282, issue.2, pp.487-502, 1984.
DOI : 10.1090/S0002-9947-1984-0732102-X

G. Michael, H. Crandall, P. Ishii, and . Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), vol.27, issue.1, pp.1-67, 1992.

G. Michael, P. Crandall, and . Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc, vol.277, issue.1, pp.1-42, 1983.

C. M. Elliott, Y. Giga, and S. Goto, Dynamic Boundary Conditions for Hamilton--Jacobi Equations, SIAM Journal on Mathematical Analysis, vol.34, issue.4, pp.861-881, 2003.
DOI : 10.1137/S003614100139957X

M. I. Freidlin and A. D. , Diffusion processes on an open book and the averaging principle, Stochastic Processes and their Applications, vol.113, issue.1, pp.101-126, 2004.
DOI : 10.1016/j.spa.2004.03.009

J. Guerand, Effective nonlinear Neumann boundary conditions for 1D nonconvex Hamilton???Jacobi equations, Journal of Differential Equations, vol.263, issue.5, 2016.
DOI : 10.1016/j.jde.2017.04.015

C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case. 28 pages. Second version, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01073954

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Annales scientifiques de l' ´ Ecole Normale Supérieure, pp.357-448, 2017.
URL : https://hal.archives-ouvertes.fr/hal-00832545

C. Imbert, R. Monneau, and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, vol.19, issue.1, pp.129-166, 2013.
DOI : 10.1051/cocv/2012002
URL : https://hal.archives-ouvertes.fr/hal-00569010

H. Ishii, Perron's method for Hamilton-Jacobi equations. Duke Math, J, vol.55, issue.2, pp.369-384, 1987.

D. Marjeta-kramar-fijav?, E. Mugnolo, and . Sikolya, Variational and Semigroup Methods for Waves and Diffusion in Networks, Applied Mathematics and Optimization, vol.55, issue.2, pp.219-240, 2007.
DOI : 10.1007/s00245-006-0887-9

P. Lions, Lectures atColì ege de France, pp.2015-2016

P. Lions and P. Souganidis, Viscosity solutions for junctions: well posedness and stability. Arxiv 1608.03682, first version, 2016.

S. Oudet, Hamilton-Jacobi equations for optimal control on multidimensional junctions, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01093112

Y. V. Pokornyi and A. V. Borovskikh, Differential Equations on Networks (Geometric Graphs), Differential equations on networks, pp.691-718, 2004.
DOI : 10.1023/B:JOTH.0000012752.77290.fa

D. Schieborn, Viscosity solutions of Hamilton-Jacobi equations of eikonal type on ramified spaces, 2006.

D. Schieborn and F. Camilli, Viscosity solutions of Eikonal equations on topological networks, Calculus of Variations and Partial Differential Equations, vol.291, issue.12, pp.671-686, 2013.
DOI : 10.1007/s11537-007-0657-8
URL : https://hal.archives-ouvertes.fr/hal-00724761

J. Von and B. , Classical solvability of linear parabolic equations on networks, J. Differential Equations, vol.72, issue.2, pp.316-337, 1988.

J. Von and B. , A maximum principle for semilinear parabolic network equations, Differential equations with applications in biology, physics, and engineering (Leibnitz, pp.37-45, 1989.

J. Von and B. , An existence result for semilinear parabolic network equations with dynamical node conditions In Progress in partial differential equations: elliptic and parabolic problems (Pont-` a- Mousson, Pitman Res. Notes Math. Ser. Longman Sci. Tech, vol.266, pp.274-283, 1991.

J. Von, B. , and S. Nicaise, Dynamical interface transition in ramified media with diffusion, Comm. Partial Differential Equations, vol.21, issue.12, pp.255-279, 1996.