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Periodic homogenisation of certain fully nonlinear partial differential equations

Published online by Cambridge University Press:  14 November 2011

Lawrence C. Evans
Lawrence C. Evans
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A.
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Synopsis

We demonstrate how a fairly simple “perturbed test function” method establishes periodic homogenisation for certain Hamilton-Jacobi and fully nonlinear elliptic partial differential equations. The idea, following Lions, Papanicolaou and Varadhan, is to introduce (possibly nonsmooth) correctors, and to modify appropriately the theory of viscosity solutions, so as to eliminate then the effects of high-frequency oscillations in the coefficients.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 3-4 , 1992 , pp. 245 - 265
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Barles, 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
2Bensoussan, 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
3Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Asymptotic Analysis for Periodic Structures (Amsterdam: North-Holland, 1978).Google Scholar
4Boccardo, L. and Murat, F.. Homogeneisation de problèmes quasi-lineaires. In Atti del Convegno Studio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna: Pitagora, 1982).Google Scholar
5Caffarelli, L.. A note on Harnack's inequality for viscosity solutions of second-order equations(preprint).Google Scholar
6Crandall, M. G., Evans, L. C. and Lions, P. L.. Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984), 487–502.CrossRefGoogle Scholar
7Crandall, M. G. and Lions, P. L.. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1–42.Google Scholar
8Weinan, E. Homogenization of conservation laws (preprint, UCLA, 1988).Google Scholar
9Evans, L. C.. The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A (to appear).Google Scholar
10Ishii, H.. On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE's. Comm. Pure Appl. Math. 42 (1989), 15–45.CrossRefGoogle Scholar
11Ishii, H.. A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations (preprint).Google Scholar
12Ishii, H. and Lions, P. L.. Viscosity solutions of fully nonlinear elliptic partial differential equations. J. Differential Equations (to appear).Google Scholar
13Jensen, R.. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal. 101 (1988), 1–27.Google Scholar
14Jensen, R.. Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations (preprint).Google Scholar
15Lions, P. L., Papanicolaou, G. and Varadhan, S.. Homogenization of Hamilton-Jacobi equations (unpublished).Google Scholar
16Lions, P. L.. Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I. Comm. Partial Differential Equations 81 (1983), 1101–1134.Google Scholar
17Tartar, L.. Cours Peccot (Collège de France, February, 1977).Google Scholar
18Trudinger, N. S.. Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations (preprint).Google Scholar
19Trudinger, N. S.. On regularity and existence of viscosity solutions of nonlinear second order elliptic equations (preprint).Google Scholar