A short proof of the C0,α-regularity of viscosity subsolutions for superquadratic viscous HamiltonJacobi equations and applications

Nonlinear Analysis, Theory, Methods and Applications

Volumn 73, Issue 1, 2010, Pages 31-47

  Barles, Guy ^a  

^a UNIVERSITÉ DE TOURS   (France)

Author keywords

Ergodic problems; Fully nonlinear equations; Hlder regularity; Homogenization; Viscosity solutions

Indexed keywords

ELLIPTIC EQUATIONS; ERGODICS; FULLY NONLINEAR EQUATIONS; HAMILTON JACOBI EQUATIONS; VISCOSITY SOLUTIONS;

VISCOSITY;

NONLINEAR EQUATIONS;

EID: 77955420247     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2010.02.009     Document Type: Article

References (39)

1. I. Capuzzo Dolcetta, F. Leoni, A. Porretta, Hlder estimates for degenerate elliptic equations with coercive Hamiltonians, Trans. Amer. Math. Soc. in press (http://www.mathaware.org/cgi-bin/mstrack/acceptedpapers/tran ).
2. J.-M. Lasry, and P.-L. Lions Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem Math. Ann. 283 4 1989 583 630
3. P.L. Lions Generalized Solutions of HamiltonJacobi Equations 1982 Pitman Boston
4. Guy Barles, and Francesca Da Lio On the generalized Dirichlet problem for viscous HamiltonJacobi equations J. Math. Pures Appl. (9) 83 1 2004 53 75
5. G. Barles, and B. Perthame Comparison principle for Dirichlet-type HamiltonJacobi equations and singular perturbations of degenerated elliptic equations Appl. Math. Optim. 21 1 1990 21 44
6. G. Barles, and B. Perthame Exit time problems in optimal control and vanishing viscosity method SIAM J. Control Optim. 26 5 1988 1133 1148
7. Guy Barles, and Francesca Da Lio Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations Adv. Differential Equations 8 8 2003 897 922
8. G. Barles, E. Rouy, and P.E. Souganidis Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations W.M. Mc Eneaney, G.G. Yin, Q. Zhang, Stochastic Analysis, Control, Optimization and Applications, (A Volume in Honnor of W.H. Fleming) 1999 Birkhuser
9. Francesca Da Lio Comparison results for quasilinear equations in annular domains and applications Comm. Partial Differential Equations 27 12 2002 283 323
10. H. Ishii Perron's method for HamiltonJacobi equations Duke Math. J. 55 2 1987 369 384
11. M.G. Crandall, H. Ishii, and P.-L. Lions User's guide to viscosity solutions of second order Partial differential equations Bull. Amer. Math. Soc. 27 1992 1 67
12. T. Tabet Tchamba, Large time behavior of solutions of viscous HamiltonJacobi Equations with superquadratic Hamiltonian. Preprint.
13. M. Bardi, and F. Da Lio Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. I: Convex operators Nonlinear Anal. Ser. A: TMA 44 8 2001 991 1006
14. M. Bardi, and F. Da Lio Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. II: Concave operators Indiana Univ. Math. J. 52 2003 607 628
15. Francesca Da Lio Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations Commun. Pure Appl. Anal. 3 3 2004 395 415
16. Lawrence C. Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc. Roy. Soc. Edinburgh Sect. A 111 34 1989 359 375
17. Lawrence C. Evans Periodic homogenisation of certain fully nonlinear partial differential equations Proc. Roy. Soc. Edinburgh Sect. A 120 34 1992 245 265
18. G. Barles, F. Da Lio, P.-L. Lions, and P.E. Souganidis Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions Indiana Univ. Math. J. 57 5 2008 2355 2375
19. G. Barles Some homogenization results for non-coercive HamiltonJacobi equations Calc. Var. Partial Differential Equations 30 2007 449 466
20. O. Alvarez Homogenization of HamiltonJacobi equations in perforated sets J. Differential Equations 159 2 1999 543 577
21. O. Alvarez, and M. Bardi Viscosity solutions methods for singular perturbations in deterministic and stochastic control SIAM J. Control Optim. 40 4 20012002 1159 1188
22. O. Alvarez, and M. Bardi Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result Arch. Ration. Mech. Anal. 170 1 2003 17 61
23. O. Alvarez, and M. Bardi Ergodic problems in differential games Advances in Dynamic Game Theory Annals of the International Society of Dynamic Games, Part II vol. 9 2007 Birkhuser Boston 131 152 http://www.springerlink.com/ content/p22392j815q1xg56/
24. ∞ J. Differential Equations 183 1 2002 132 164
25. O. Alvarez, and H. Ishii HamiltonJacobi equations with partial gradient and application to homogenization Comm. Partial Differential Equations 26 56 2001 983 1002
26. M. Arisawa Ergodic problem for the HamiltonJacobiBellman equation. I. Existence of the ergodic attractor Ann. Inst. H. Poincaré Anal. Non Linaire 14 1997 415 438
27. M. Arisawa Ergodic problem for the HamiltonJacobiBellman equation. II Ann. Inst. H. Poincaré Anal. Non Linaire 15 1998 1 24
28. M. Arisawa, and P.-L. Lions On ergodic stochastic control Comm. Partial Differential Equations 23 1112 1998 2187 2217
29. Z. Artstein, and V. Gaitsgory The value function of singularly perturbed control systems Appl. Math. Optim. 41 3 2000 425 445 (Pubitemid 30596843)
30. M. Bardi On differential games with long-time-average cost Advances in dynamic games and their applications Ann. Internat. Soc. Dynam. Games vol. 10 2009 Birkhuser Boston, Inc. Boston, MA 3 18
31. G. Barles Solutions de viscosit des quations de HamiltonJacobi Collection "Mathématiques et Applications" of SMAI vol. 17 1994 Springer-Verlag
32. G. Barles, S. Biton, and O. Ley A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations Arch. Ration. Mech. Anal. 162 4 2002 287 325
33. G. Barles, and P.E. Souganidis Spacetime periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations SIAM J. Math. Anal. 32 6 2001 1311 1323
34. C. Imbert, and R. Monneau Homogenization of first-order equations with u ε -periodic Hamiltonians. Part I. Local equations Arch. Ration. Mech. Anal. 187 1 2008 49 89
35. K. Horie, and H. Ishii Homogenization of HamiltonJacobi equations on domains with small scale periodic structure Indiana Univ. Math. J. 47 3 1998 1011 1058
36. H. Ishii Homogenization of the Cauchy problem for HamiltonJacobi equations Stochastic Analysis, Control, Optimization and Applications Systems Control Found. Appl. 1999 Birkhuser Boston Boston, MA 305 324
37. H. Ishii Almost periodic homogenization of HamiltonJacobi equations International Conference on Differential Equations, Vol. 1, 2 Berlin, 1999 2000 World Sci. Publishing River Edge, NJ 600 605
38. P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan, Homogenization of HamiltonJacobi Equations, unpublished work.
39. P.E. Souganidis Stochastic homogenization of HamiltonJacobi equations and some applications Asymptot. Anal. 20 1 1999 1 11