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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 111, Issue 3-4
  • January 1989, pp. 359-375

The perturbed test function method for viscosity solutions of nonlinear PDE

  • Lawrence C. Evans (a1)
  • DOI: http://dx.doi.org/10.1017/S0308210500018631
  • Published online: 14 November 2011
Abstract
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.

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2G. Barles and B. Perthame . Exit time problems in optimal control and the vanishing viscosity method. SIAM J. Control Optim. 26 (1988), 11331148.

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7M. G. Crandall and P. L. Lions . Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 142.

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12N. Fusco and Moscariello . On homogenization of quasilinear divergence structure operators. Ann. Mat. Pura Appl. 146 (1987), 113.

14D. Gilbarg and N. S. Trudinger . Elliptic Partial Differential Equations of Second Order, 2nd edn (Berlin: Springer, 1983).

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21G. Papanicolaou and S. R. S. Varadhan . A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations. Comm. Pure Appl. Math. 26 (1973), 497524.

22M. Pinsky . 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), 101111.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
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