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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 $F\left(x,u,\nabla u,{D}^{2}u\right)=0$ in ${\Omega }$ 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

$u-v\le \underset{x\in \partial {\Omega }}{sup}{\left\{{u}^{*}\left(x\right)-{v}_{*}\left(x\right)\right\}}^{+}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }$

${sup}_{y\in {\Omega },y\to x}supu\left(y\right)$ and ${v}_{*}=-{\left(-v\right)}^{*}·$

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 ${C}_{1,\alpha }$-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:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B50 Maximum principles (PDE) 35B65 Smoothness and regularity of solutions of PDE 35J70 Degenerate elliptic equations 35D10 Regularity of generalized solutions of PDE (MSC2000)