Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

The perturbed test function method for viscosity solutions of nonlinear PDE

Lawrence C. Evansa1 p1

a1 Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.

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.

(Received July 21 1988)

(Revised November 07 1988)

Correspondence

p1 Supported in part by NSF grant DMS-86-01532. The author was a part-time member of the Institute for Physical Science and Technology, University of Maryland, when this paper was written