This is a long and technical paper about viscosity solutions for fully nonlinear elliptic equations in under various boundary conditions. The common strategy to tackle these equations is the observation of the first author [Duke Math. J. 55, 362- 384 (1987) and Commun. Pure Appl. Math. 42, No., 15-45 (1989; Zbl 0645.35025)] that (unique) existence is implied by a Perron-process, if viscosity sub- and supersolutions are known and a kind of maximum principle can be proved. It reads as follows: Whenever u (resp. v) is an usc (resp. lsc) bounded viscosity sub- (resp. super-) solution, then
Hence the problem remains in (and most of the paper is devoted to) veryfying this under various structure conditions on F, including Isaac- Bellman equations and also Monge-Ampère equations. See also R. Jensen [Arch. Ration. Mech. Anal. 101, No.1, 1-27 (1988)]. The paper closes with some remarks to the regularity of solutions. For -estimates, see also N. S. Trudinger [Proc. R. Soc. Edinb. Sect. A 108, No.1/2, 57-65 (1988; Zbl 0653.35026)].