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Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. (English) Zbl 0708.35031

This is a long and technical paper about viscosity solutions for fully nonlinear elliptic equations F(x,u,u,D 2 u)=0 in Ω 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

u-vsup xΩ {u * (x)-v * (x)} + inΩ

sup yΩ,yx supu(y) and v * =-(-v) * ·

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 C 1,α -estimates, see also N. S. Trudinger [Proc. R. Soc. Edinb. Sect. A 108, No.1/2, 57-65 (1988; Zbl 0653.35026)].

Reviewer: M.Wiegner
MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35B50Maximum principles (PDE)
35B65Smoothness and regularity of solutions of PDE
35J70Degenerate elliptic equations
35D10Regularity of generalized solutions of PDE (MSC2000)