Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T00:16:30.907Z Has data issue: false hasContentIssue false
  • English
  • Français

The perturbed test function method for viscosity solutions of nonlinear PDE

Published online by Cambridge University Press:  14 November 2011

Lawrence C. Evans
Lawrence C. Evans
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

The method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 111 , Issue 3-4 , 1989 , pp. 359 - 375
Copyright
Copyright © Royal Society of Edinburgh 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Attouch, H.. Variational Convergence for Functions and Operators (New York: Pitman, 1984).Google Scholar
2Barles, G. and Perthame, B.. Exit time problems in optimal control and the vanishing viscosity method. SIAM J. Control Optim. 26 (1988), 1133–1148.Google Scholar
3Bensoussan, A.. Methodes de Perturbations en Contrôle Optimal (to appear).Google Scholar
4Bensoussan, A., Boccardo, L. and Murat, F.. Homogenization of elliptic equations with principal part not in divergence form and Hamiltonian with quadratic growth. Comm. Pure Appl. Math. 39 (1986), 769–805.Google Scholar
5Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Asymptotic Analysis for Periodic Structures (Amsterdam: North Holland, 1978).Google Scholar
6Boccardo, L. and Murat, F.. Homogenisation de problemes quasi-linearies. In Studio di Problemi-Limite delta Analisi Funzionale, 13–51 (Bologna: Pitagora Editrice, 1982).Google Scholar
7Crandall, M. G. and Lions, P. L.. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1–42.Google Scholar
8Crandall, M. G. and Sougandis, P. E.. Developments in the theory of nonlinear first order partial differential equations. In Proc. International Sym. on Diff. Eq. (Amsterdam: North Holland, 1984).Google Scholar
9Evans, L. C.. A convergence theorem for solutions of nonlinear second order elliptic equations. Indiana Univ. Math. J. 27 (1978), 875–887.Google Scholar
10Evans, L. C.. Nonlinear semigroup theory and viscosity solutions of Hamilton-Jacobi PDE. In Nonlinear Semigroups, Partial Differential Equations and Attractors, eds. Gill, T. L. and Zachary, W. W., Lecture Notes in Mathematics 1248 (Berlin: Springer, 1987).CrossRefGoogle Scholar
11Evans, L. C. and Lions, P. L., (to appear).Google Scholar
12Fusco, N. and , Moscariello. On homogenization of quasilinear divergence structure operators. Ann. Mat. Pura Appl. 146 (1987), 1–13.CrossRefGoogle Scholar
13Gantmacher, F. R.. The Theory of Matrices Vol. II (New York: Chelsea, 1960).Google Scholar
14Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn (Berlin: Springer, 1983).Google Scholar
15Ishii, H.. A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations (to appear).Google Scholar
16Ishii, H.. On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE's (to appear).Google Scholar
17Jensen, R.. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal. 101 (1988), 1–27.CrossRefGoogle Scholar
18Kushner, H.. Approximation and Weak Convergence Methods for Random Processes. (Cambridge: MIT Press, 1984).Google Scholar
19Lions, P. L.. Generalized Solutions of Hamilton-Jacobi Equations (Boston: Pitman, 1982).Google Scholar
20Lions, P. L., Papanicolaou, G. and Varadhan, S. R. S.. Homogenization of Hamilton-Jacobi equations (preprint).Google Scholar
21Papanicolaou, G. and Varadhan, S. R. S.. A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations. Comm. Pure Appl. Math. 26 (1973), 497–524.CrossRefGoogle Scholar
22Pinsky, M.. Differential equations with a small parameter and the central limit theorem for functions denned on a finite Markov chain. Z. Wahrsch. Verw. Gebiete 9 (1968), 101–111.CrossRefGoogle Scholar
23Tartar, L.. Cours Peccot, Collège de France, February, 1977.Google Scholar