The perturbed test function method for viscosity solutions of nonlinear PDE. (English) Zbl 0679.35001

Evans, Lawrence C.

Proc. R. Soc. Edinb., Sect. A 111, No. 3-4, 359-375 (1989).

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

Keywords:

perturbed test function method; maximum principle; viscosity solutions; nonlinear first and second order partial differential equations; Hamilton-Jacobi PDEs

Citations:

Zbl 0599.35024

References:

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