*(English)*Zbl 0695.35007

The author considers fully nonlinear elliptic differential equations of second order:

Assuming, that $F$ is uniformly elliptic, Lipschitz-continuous with respect to $Du$, monotone in $u$ and uniformly continuous in $x$, he shows the following comparison principle:

Let $u,v\in {C}^{0}\left(\overline{{\Omega}}\right)\cap {C}^{0,1}\left({\Omega}\right)$ be respectively viscosity the subsolution and supersolution of (*) in ${\Omega}$ with $u\le v$ on $\partial {\Omega}$. Then we have $u\le v$ in ${\Omega}$.

First he proves the theorem under stronger continuity assumptions on $F$. Following an idea of *R. Jensen* [Arch. Ration. Mech. Anal. 101, 1–27 (1988; Zbl 0708.35019)], $u$ and $v$ 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 $F$ with respect to $x$, 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.

##### MSC:

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. (PDE) |

35J65 | Nonlinear boundary value problems for linear elliptic equations |