The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations Authors Robert Jensen Department of Mathematical Sciences Loyola University of Chicago Article

DOI :
10.1007/BF00281780

Cite this article as: Jensen, R. Arch. Rational Mech. Anal. (1988) 101: 1. doi:10.1007/BF00281780
Abstract We prove that viscosity solutions in W ^{1,∞} of the second order, fully nonlinear, equation F (D ^{2} u , Du, u ) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u . We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D ^{2} u and from that used by Crandall & Lions [3] and Crandall , Evans & Lions [2] for fully nonlinear first order problems.

Communicated by C. M. Dafermos

The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.

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