On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions
Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 5, pp. 521-541. 
@article{AIHPC_2005__22_5_521_0,
     author = {Barles, Guy and Da Lio, Francesca},
     title = {On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear {Neumann} boundary conditions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {521--541},
     publisher = {Elsevier},
     volume = {22},
     number = {5},
     year = {2005},
     doi = {10.1016/j.anihpc.2004.09.001},
     mrnumber = {2171989},
     zbl = {02235966},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2004.09.001/}
}
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Barles, Guy; Da Lio, Francesca. On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions. Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 5, pp. 521-541. doi : 10.1016/j.anihpc.2004.09.001. https://www.numdam.org/articles/10.1016/j.anihpc.2004.09.001/

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