Springer Online Journal Archives 1860-2000
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  and Crandall, Evans & Lions  for fully nonlinear first order problems.
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