Abstract
The main result of the paper is a general convergence theorem for the viscosity solutions of singular perturbation problems for fully nonlinear degenerate parabolic PDEs (partial differential equations) with highly oscillating initial data. It substantially generalizes some results obtained previously in [2]. Under the only assumptions that the Hamiltonian is ergodic and stabilizing in a suitable sense, the solutions are proved to converge in a relaxed sense to the solution of a limit Cauchy problem with appropriate effective Hamiltonian and initial data. In its formulation, our convergence result is analogous to the stability property of Barles and Perthame. It should thus reveal a useful tool for studying general singular perturbation problems by viscosity solutions techniques. A detailed exposition of ergodicity and stabilization is given, with many examples. Applications to homogenization and averaging are also discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alvarez, O.: Homogenization of Hamilton-Jacobi equations in perforated sets. J. Differential Equations 159, 543–577 (1999)
Alvarez, O., Bardi, M.: Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40, 1159–1188 (2001)
Alvarez, O., Bardi, M.: Ergodicity, stabilization and singular perturbations for Bellman-Isaacs equations. University of Padova, Preprint, 2003
Alvarez, O., Bardi, M.: Homogenization of fully nonlinear PDEs with periodically oscillating data. University of Padova, Preprint, 2003
Arisawa, M.: Ergodic problem for the Hamilton-Jacobi-Bellman equation II. Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 1–24 (1998)
Arisawa, M., Lions, P.-L.: On ergodic stochastic control. Comm. Partial Differential Equations 23, 2187–2217 (1998)
Arnold, V.I., Avez, A.: Problèmes ergodiques de la mécanique classique. Gauthier-Villars, Paris, 1967. English translation: Benjamin, W. A., New York, 1968
Artstein, Z., Gaitsgory, V.: The value function of singularly perturbed control systems. Appl. Math. Optim. 41, 425–445 (2000)
Bagagiolo, F., Bardi, M.: Singular perturbation of a finite horizon problem with state-space constraints. SIAM J. Control Optim. 36, 2040–2060 (1998)
Bardi, M.: Homogenization of quasilinear elliptic equations with possibly superquadratic growth. In: Marino, A., Murthy, M. K. V. (eds), Nonlinear variational problems and partial differential equations (Isola d’Elba, 1990), number 320 in Pitman Res. Notes Math., Longman, Harlow, pp. 45–56 1995
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston, 1997
Barles, G.: Solutions de Viscosité des Equations de Hamilton-Jacobi. Number~17 in Mathématiques et Applications. Springer-Verlag, Paris, 1994
Barles, G., Souganidis, P.E.: On the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 31, 925–939 (2000)
Barron, E.N.: Averaging in Lagrange and minimax problems of optimal control. SIAM J. Control Optim. 31, 1630–1652 (1993)
Bensoussan, A.: Perturbation methods in optimal control. Wiley/Gauthiers-Villars, Chichester, 1988
Bensoussan, A., Boccardo, L., Murat, F.: Homogenization of elliptic equations with principal part not in divergence form and Hamiltonian with quadratic growth. Comm. Pure Appl. Math. 39, 769–805 (1986)
Bensoussan, A., Lions, J.-L., Papanicolaou G.: Asymptotic Analysis for periodic Structures. North-Holland, Amsterdam, 1978
Brahim-Otsmane, S., Francfort, G.A., Murat, F.: Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 71, 197–231 (1992)
Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic theory. Springer-Verlag, Berlin, 1982
Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27, 1–67 (1992)
Evans, L.C.: The perturbed test function method for viscosity solutions of nonlinear P.D.E. Proc. Roy. Soc. Edinburgh Sect. A 111, 359–375 (1989)
Evans, L.C.: Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 120, 245–265 (1992)
Evans, L.C., Gomes, D.: Effective Hamiltonians and averaging for hamiltonian dynamics. I. Arch. Rational Mech. Anal. 157, 1–33 (2001)
Evans, L.C., Gomes, D.: Effective Hamiltonians and averaging for hamiltonian dynamics. II. Arch. Rational Mech. Anal. 161, 271–305 (2002)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, Berlin, 1993
Has’minskiĭ, R.Z.: Stochastic stability of differential equations. Sijthoff, Noordhoff, Alphen aan den Rijn, 1980
Ishii, H.: On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde’s. Comm. Pure Appl. Math. 42, 15–45 (1989)
Ishii, H.: Homogenization of the Cauchy problem for Hamilton-Jacobi equations. In: McEneaney, W. M., Yin, G., Zhang, Q., (eds), Stochastic analysis, control, optimization and applications. A volume in honor of Wendell H. Fleming, Birkhäuser, Boston, pp. 305–324 1999
Jensen, R., Lions, P.-L.: Some asymptotic problems in fully nonlinear elliptic equations and stochastic control. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11, 129–176 (1984)
Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994
Kokotović, P.V., Khalil, H.K., O’Reilly, J.: Singular perturbation methods in control: analysis and design. Academic Press, London, 1986
Kushner, H.J.: Weak convergence methods and singularly perturbed stochastic control and filtering problems. Birkhäuser, Boston, 1990
Lasry, J.-M., Lions, P.-L.: Une classe nouvelle de problèmes singuliers de contrôle stochastique [A new class of singular stochastic control problems]. C. R. Acad. Sci. Paris Sér. I Math. 331, 879–885 (2000)
Lions, P.-L.: Generalized solutions of Hamilton-Jacobi equations. Pitman, Boston, 1982
Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogeneization of Hamilton-Jacobi equations. Unpublished, 1986
Rezakhanlou, F., Tarver, J.E.: Homogenization for stochastic Hamilton-Jacobi equations. Arch. Rational Mech. Anal. 151, 277–309 (2000)
Simon, B.: Functional integration and quantum physics. Academic Press, New York, 1979
Souganidis, P.E.: Stochastic homogenization of Hamilton-Jacobi equations and some applications. Asymptotic Anal. 20, 1–11 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. C. Evans
Rights and permissions
About this article
Cite this article
Alvarez, O., Bardi, M. Singular Perturbations of Nonlinear Degenerate Parabolic PDEs: a General Convergence Result. Arch. Rational Mech. Anal. 170, 17–61 (2003). https://doi.org/10.1007/s00205-003-0266-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-003-0266-5