On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions

Guy Barles; Francesca Da Lio

Annales de l'I.H.P. Analyse non linéaire (2005)

  • Volume: 22, Issue: 5, page 521-541
  • ISSN: 0294-1449

How to cite

top

Barles, Guy, and 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 22.5 (2005): 521-541. <http://eudml.org/doc/78667>.

@article{Barles2005,
author = {Barles, Guy, Da Lio, Francesca},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {fully nonlinear elliptic equation},
language = {eng},
number = {5},
pages = {521-541},
publisher = {Elsevier},
title = {On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions},
url = {http://eudml.org/doc/78667},
volume = {22},
year = {2005},
}

TY - JOUR
AU - Barles, Guy
AU - Da Lio, Francesca
TI - On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2005
PB - Elsevier
VL - 22
IS - 5
SP - 521
EP - 541
LA - eng
KW - fully nonlinear elliptic equation
UR - http://eudml.org/doc/78667
ER -

References

top
  1. [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. Zbl1017.49028MR1882729
  2. [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. Zbl1032.35103MR2012646
  3. [3] Arisawa M., Ergodic problem for the Hamilton–Jacobi–Bellman equation. I. Existence of the ergodic attractor, Ann. Inst. H. Poincaré Anal. Non Linéaire14 (4) (1997) 415-438. Zbl0892.49015MR1464529
  4. [4] Arisawa M., Ergodic problem for the Hamilton–Jacobi–Bellman equation. II, Ann. Inst. Poincaré Anal. Non Linéaire15 (1) (1998) 1-24. Zbl0903.49018MR1614615
  5. [5] Arisawa M., Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Linéaire20 (2) (2003) 293-332. Zbl1139.35311MR1961518
  6. [6] Arisawa M., Lions P.-L., On ergodic stochastic control, Comm. Partial Differential Equations23 (11–12) (1998) 2187-2217. Zbl1126.93434MR1662180
  7. [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. Zbl0881.35013MR1452882
  8. [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. Zbl0939.35038MR1710100
  9. [9] Barles G., Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications, J. Differential Equations154 (1999) 191-224. Zbl0924.35051MR1685618
  10. [10] G. Barles, F. Da Lio, Local C 0 , α estimates for viscosity solutions of Neumann-type boundary value problems, preprint. Zbl1147.35312MR2228697
  11. [11] Barles G., Lions P.L., Remarques sur les problèmes de riflexion obliques, C. R. Acad. Sci. Paris, Ser. I320 (1995) 69-74. Zbl0831.60068MR1320834
  12. [12] G. Barles, M. Ramaswamy, Sufficient structure conditions for uniqueness of viscosity solutions of semilinear and quasilinear equations, NoDEA, in press. Zbl1146.35326MR2199386
  13. [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. Zbl0960.70015MR1752423
  14. [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. Zbl0986.35047MR1856250
  15. [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. Zbl0648.49001MR949208
  16. [16] Bensoussan A., Frehse J., On Bellman equations of ergodic control in R n , J. Reine Angew. Math.429 (1992) 125-160. Zbl0779.35038MR1173120
  17. [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. Zbl1050.93075MR1931643
  18. [18] Capuzzo Dolcetta I., Lions P.L., Hamilton–Jacobi equations with state constraints, Trans. Amer. Math. Soc.318 (2) (1990) 643-683. Zbl0702.49019MR951880
  19. [19] Concordel M.C., Periodic homogenization of Hamilton–Jacobi equations: additive eigenvalues and variational formula, Indiana Univ. Math. J.45 (4) (1996) 1095-1117. Zbl0871.49025MR1444479
  20. [20] Concordel M.C., Periodic homogenization of Hamilton–Jacobi equations: II: Eikonal equations, Proc. Roy. Soc. Edinburgh Sect. A127 (4) (1997) 665-689. Zbl0883.35010MR1465414
  21. [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. Zbl0755.35015MR1118699
  22. [22] Evans L.C., The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A111 (3–4) (1989) 359-375. Zbl0679.35001MR1007533
  23. [23] Evans L.C., Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A120 (3–4) (1992) 245-265. Zbl0796.35011MR1159184
  24. [24] Evans L.C., Gomes D., Effective Hamiltonians and averaging for Hamiltonian dynamics. I., Arch. Rational Mech. Anal.157 (1) (2001) 1-33. Zbl0986.37056MR1822413
  25. [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. Zbl0885.58022MR1451248
  26. [26] Fathi A., Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris, Sér. I Math.325 (6) (1997) 649-652. Zbl0943.37031MR1473840
  27. [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. Zbl1052.37514MR1650261
  28. [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. Zbl0969.35018MR1870203
  29. [29] Ishii H., Perron's method for Hamilton–Jacobi equations, Duke Math. J.55 (1987) 369-384. Zbl0697.35030MR894587
  30. [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. Zbl0688.49026MR990591
  31. [31] Lions P.-L., Neumann type boundary conditions for Hamilton–Jacobi equations, Duke Math. J.52 (4) (1985) 793-820. Zbl0599.35025MR816386
  32. [32] P.-L. Lions, G. Papanicolaou, S.R.S Varadhan, unpublished preprint. 
  33. [33] Lions P.L., Sznitman A.S., Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math.XXXVII (1984) 511-537. Zbl0598.60060MR745330
  34. [34] Namah G., Roquejoffre J.-M., Remarks on the long time behaviour of the solutions of Hamilton–Jacobi equations, Comm. Partial Differential Equations24 (5–6) (1999) 883-893. Zbl0924.35028MR1680905

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.