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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: $F(x,u,Du,D^ 2u)=0. \tag{*}$ 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 $$u,v\in C^ 0({\bar \Omega})\cap C^{0,1}(\Omega)$$ be respectively viscosity the subsolution and supersolution of (*) in $$\Omega$$ with $$u\leq v$$ on $$\partial \Omega$$. Then we have $$u\leq v$$ 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. in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
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