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Periodic homogenisation of certain fully nonlinear partial differential equations

  • Lawrence C. Evans (a1)

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.

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1Barles, G. and Perthame, B.. Exit time problems in optimal control and the vanishing viscosity method. SIAM J. Control Optim. 26 (1988), 11331148.
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), 769805.
3Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Asymptotic Analysis for Periodic Structures (Amsterdam: North-Holland, 1978).
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).
5Caffarelli, L.. A note on Harnack's inequality for viscosity solutions of second-order equations(preprint).
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), 487502.
7Crandall, M. G. and Lions, P. L.. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 142.
8Weinan, E. Homogenization of conservation laws (preprint, UCLA, 1988).
9Evans, L. C.. The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A (to appear).
10Ishii, H.. On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE's. Comm. Pure Appl. Math. 42 (1989), 1545.
11Ishii, H.. A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations (preprint).
12Ishii, H. and Lions, P. L.. Viscosity solutions of fully nonlinear elliptic partial differential equations. J. Differential Equations (to appear).
13Jensen, R.. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal. 101 (1988), 127.
14Jensen, R.. Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations (preprint).
15Lions, P. L., Papanicolaou, G. and Varadhan, S.. Homogenization of Hamilton-Jacobi equations (unpublished).
16Lions, P. L.. Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I. Comm. Partial Differential Equations 81 (1983), 11011134.
17Tartar, L.. Cours Peccot (Collège de France, February, 1977).
18Trudinger, N. S.. Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations (preprint).
19Trudinger, N. S.. On regularity and existence of viscosity solutions of nonlinear second order elliptic equations (preprint).
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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