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The perturbed test function method for viscosity solutions of nonlinear PDE. (English) Zbl 0679.35001
In this carefully written, largely expository paper the author shows how to combine the perturbed test function method with maximum principle arguments related to viscosity solutions for nonlinear first and second order partial differential equations. This method replaces the fixed smooth test function used in the theory of viscosity solutions [see M. G. Crandall and P.-L. Lions, Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)] with modified test functions which contain lower order correctors. Such techniques have already been successful in probability and in the calculus of variations. The author’s first example is homogenization for quasilinear elliptic PDEs with highly oscillatory, periodic coefficients. The second is concerned with the convergence of a system of first order Hamilton-Jacobi PDEs to a single, second order quasilinear parabolic PDE.
Reviewer: Simeon Reich

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35F20 Nonlinear first-order PDEs
35J60 Nonlinear elliptic equations
35B20 Perturbations in context of PDEs
35G20 Nonlinear higher-order PDEs
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References:
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