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Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations. (English) Zbl 0695.35007

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

$F\left(x,u,Du,{D}^{2}u\right)=0·\phantom{\rule{2.em}{0ex}}\left(*\right)$

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.

Reviewer: H.-Ch. Grunau

MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35J65 Nonlinear boundary value problems for linear elliptic equations