In the framework of viscosity solutions, we give an extension of the strong comparison result for Hamilton–Jacobi–Bellman (HJB) equations with Dirichlet boundary conditions to the case of some non-smooth domains. In particular, it may be applied to parabolic problems on cylindrical domains.
Dans le cadre de la théorie des solutions de viscosité, on donne une extension du principe de comparaison fort pour l'équation d'Hamilton–Jacobi–Bellman (HJB) avec condition au bord de type Dirichlet au cas de certains domaines irréguliers. En particulier, ce résultat est applicable aux problèmes paraboliques posés dans des domaines cylindriques.
Accepted:
Published online:
Sébastien Chaumont 1
@article{CRMATH_2004__339_8_555_0, author = {S\'ebastien Chaumont}, title = {Uniqueness to elliptic and parabolic {Hamilton{\textendash}Jacobi{\textendash}Bellman} equations with non-smooth boundary}, journal = {Comptes Rendus. Math\'ematique}, pages = {555--560}, publisher = {Elsevier}, volume = {339}, number = {8}, year = {2004}, doi = {10.1016/j.crma.2004.08.009}, language = {en}, }
TY - JOUR AU - Sébastien Chaumont TI - Uniqueness to elliptic and parabolic Hamilton–Jacobi–Bellman equations with non-smooth boundary JO - Comptes Rendus. Mathématique PY - 2004 SP - 555 EP - 560 VL - 339 IS - 8 PB - Elsevier DO - 10.1016/j.crma.2004.08.009 LA - en ID - CRMATH_2004__339_8_555_0 ER -
Sébastien Chaumont. Uniqueness to elliptic and parabolic Hamilton–Jacobi–Bellman equations with non-smooth boundary. Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 555-560. doi : 10.1016/j.crma.2004.08.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.009/
[1] Solutions de Viscosité des Equations de Hamilton–Jacobi, Math. Appl., vol. 17, Springer-Verlag, 1994
[2] Nonlinear Neumann boundary conditions for quasilinear elliptic equations and applications, J. Differential Equations, Volume 154 (1999) no. 1, pp. 191-224
[3] Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep., Volume 60 (1997), pp. 57-83
[4] The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems, Comm. Partial Differential Equations, Volume 20 (1995) no. 1/2, pp. 129-178
[5] A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its applications, Comm. Partial Differential Equations, Volume 23 (1998) no. 11/12, pp. 1945-2033
[6] Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., vol. 4, Elsevier, North-Holland, 1991, pp. 273-283
[7] User's guide to viscosity solutions of 2nd order PDE, Bull. Amer. Math. Soc., Volume 27 (1992) no. 1, pp. 1-67
[8] Viscosity solutions of second order fully nonlinear elliptic equations with state constraints, Indiana Univ. Math. J., Volume 43 (1994) no. 2, pp. 493-517
[9] Controlled Diffusion Processes, Springer-Verlag, Berlin, 1980
[10] Optimal control of diffusion processes and HJB equations part II: viscosity solutions and uniqueness, Comm. Partial Differential Equations, Volume 8 (1983), pp. 1229-1276
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