On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions
Barles, Guy ; Da Lio, Francesca
Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005), p. 521-541 / Harvested from Numdam
@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},
     volume = {22},
     year = {2005},
     pages = {521-541},
     doi = {10.1016/j.anihpc.2004.09.001},
     mrnumber = {2171989},
     zbl = {02235966},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2005__22_5_521_0}
}
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) pp. 521-541. doi : 10.1016/j.anihpc.2004.09.001. http://gdmltest.u-ga.fr/item/AIHPC_2005__22_5_521_0/

[1] Alvarez O., Bardi M., Viscosity solutions methods for singular perturbations in deterministic and stochastic control, SIAM J. Control Optim. 40 (4) (2001/02) 1159-1188. | MR 1882729 | Zbl 1017.49028

[2] Alvarez O., Bardi M., Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result, Arch. Rational Mech. Anal. 170 (1) (2003) 17-61. | MR 2012646 | Zbl 1032.35103

[3] Arisawa M., Ergodic problem for the Hamilton-Jacobi-Bellman equation. I. Existence of the ergodic attractor, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (4) (1997) 415-438. | Numdam | MR 1464529 | Zbl 0892.49015

[4] Arisawa M., Ergodic problem for the Hamilton-Jacobi-Bellman equation. II, Ann. Inst. Poincaré Anal. Non Linéaire 15 (1) (1998) 1-24. | Numdam | MR 1614615 | Zbl 0903.49018

[5] Arisawa M., Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Linéaire 20 (2) (2003) 293-332. | Numdam | MR 1961518 | Zbl 1139.35311 | Zbl 01912452

[6] Arisawa M., Lions P.-L., On ergodic stochastic control, Comm. Partial Differential Equations 23 (11-12) (1998) 2187-2217. | MR 1662180 | Zbl 1126.93434 | Zbl 01247980

[7] Bagagiolo F., Bardi M., Capuzzo Dolcetta I., A viscosity solutions approach to some asymptotic problems in optimal control, in: Partial Differential Equation Methods in Control and Shape Analysis (Pisa), Lecture Notes in Pure and Appl. Math., vol. 188, Dekker, New York, 1997, pp. 29-39. | MR 1452882 | Zbl 0881.35013

[8] Bardi M., Da Lio F., On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math. (Basel) 73 (4) (1999) 276-285. | MR 1710100 | Zbl 0939.35038

[9] Barles G., Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications, J. Differential Equations 154 (1999) 191-224. | MR 1685618 | Zbl 0924.35051

[10] G. Barles, F. Da Lio, Local C 0,α estimates for viscosity solutions of Neumann-type boundary value problems, preprint. | MR 2228697

[11] Barles G., Lions P.L., Remarques sur les problèmes de riflexion obliques, C. R. Acad. Sci. Paris, Ser. I 320 (1995) 69-74. | MR 1320834 | Zbl 0831.60068

[12] G. Barles, M. Ramaswamy, Sufficient structure conditions for uniqueness of viscosity solutions of semilinear and quasilinear equations, NoDEA, in press. | MR 2199386 | Zbl 05068179

[13] Barles G., Souganidis P.E., On the large time behaviour of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal. 31 (4) (2000) 925-939. | MR 1752423 | Zbl 0960.70015

[14] Barles G., Souganidis P.E., Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal. 32 (6) (2001) 1311-1323. | MR 1856250 | Zbl 0986.35047

[15] Bensoussan A., Perturbation methods in optimal control, Wiley/Gauthier-Villars Series in Modern Applied Mathematics, Wiley, Chichester, 1988, Translated from the French by C. Tomson. | MR 949208 | Zbl 0648.49001

[16] Bensoussan A., Frehse J., On Bellman equations of ergodic control in R n , J. Reine Angew. Math. 429 (1992) 125-160. | MR 1173120 | Zbl 0779.35038

[17] Bensoussan A., Frehse J., Ergodic control Bellman equation with Neumann boundary conditions, in: Stochastic Theory and Control (Lawrence, KS, 2001), Lecture Notes in Control and Inform. Sci., vol. 280, Springer, Berlin, 2002, pp. 59-71. | MR 1931643 | Zbl 1050.93075

[18] Capuzzo Dolcetta I., Lions P.L., Hamilton-Jacobi equations with state constraints, Trans. Amer. Math. Soc. 318 (2) (1990) 643-683. | MR 951880 | Zbl 0702.49019

[19] Concordel M.C., Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula, Indiana Univ. Math. J. 45 (4) (1996) 1095-1117. | MR 1444479 | Zbl 0871.49025

[20] Concordel M.C., Periodic homogenization of Hamilton-Jacobi equations: II: Eikonal equations, Proc. Roy. Soc. Edinburgh Sect. A 127 (4) (1997) 665-689. | MR 1465414 | Zbl 0883.35010

[21] Crandal M.G., Ishii H., Lions P.L., User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Soc. 27 (1992) 1-67. | MR 1118699 | Zbl 0755.35015

[22] Evans L.C., The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A 111 (3-4) (1989) 359-375. | MR 1007533 | Zbl 0679.35001

[23] Evans L.C., Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 120 (3-4) (1992) 245-265. | MR 1159184 | Zbl 0796.35011

[24] Evans L.C., Gomes D., Effective Hamiltonians and averaging for Hamiltonian dynamics. I., Arch. Rational Mech. Anal. 157 (1) (2001) 1-33. | MR 1822413 | Zbl 0986.37056

[25] Fathi A., Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris, Sér. I Math. 324 (1997) 1043-1046. | MR 1451248 | Zbl 0885.58022

[26] Fathi A., Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris, Sér. I Math. 325 (6) (1997) 649-652. | MR 1473840 | Zbl 0943.37031

[27] Fathi A., Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris, Sér. I Math. 327 (3) (1998) 267-270. | MR 1650261 | Zbl 1052.37514

[28] Ishii H., Almost periodic homogenization of Hamilton-Jacobi equations, in: International Conference on Differential Equations, vols. 1, 2 (Berlin, 1999), World Sci., River Edge, NJ, 2000, pp. 600-605. | MR 1870203 | Zbl 0969.35018

[29] Ishii H., Perron's method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987) 369-384. | MR 894587 | Zbl 0697.35030

[30] Lasry J.M., Lions P.L., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, Math. Ann. 283 (1989) 583-630. | MR 990591 | Zbl 0688.49026

[31] Lions P.-L., Neumann type boundary conditions for Hamilton-Jacobi equations, Duke Math. J. 52 (4) (1985) 793-820. | MR 816386 | Zbl 0599.35025

[32] P.-L. Lions, G. Papanicolaou, S.R.S Varadhan, unpublished preprint.

[33] Lions P.L., Sznitman A.S., Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. XXXVII (1984) 511-537. | MR 745330 | Zbl 0598.60060

[34] Namah G., Roquejoffre J.-M., Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations 24 (5-6) (1999) 883-893. | MR 1680905 | Zbl 0924.35028