The author considers fully nonlinear elliptic differential equations of second order:
Assuming, that is uniformly elliptic, Lipschitz-continuous with respect to , monotone in and uniformly continuous in , he shows the following comparison principle:
Let be respectively viscosity the subsolution and supersolution of (*) in with on . Then we have in .
First he proves the theorem under stronger continuity assumptions on . Following an idea of R. Jensen [Arch. Ration. Mech. Anal. 101, 1–27 (1988; Zbl 0708.35019)], and are approximated by respectively semiconvex and semiconcave functions; the difference of them is shown to satisfy a linear differential inequality. Application of the Alexandrov maximum principle and other devices of the linear theory yields the result.
To relax the continuity assumptions on with respect to , rather subtle arguments are necessary. For example, the above mentioned difference is to be modified by a convex solution of a Monge-Ampère equation.
In the last part of the paper, pointwise estimates for viscosity solutions of (*) (local maximum principle, Harnack inequality, Hölder estimate) are given.