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Zbl 0695.35007
Trudinger, Neil S.
Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations. (English)
[J] Rev. Mat. Iberoam. 4, No. 3-4, 453-468 (1988). ISSN 0213-2230

The author considers fully nonlinear elliptic differential equations of second order: $$F(x,u,Du,D\sp 2u)=0. \tag *$$ 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: \par Let $u,v\in C\sp 0({\bar \Omega})\cap C\sp{0,1}(\Omega)$ be respectively viscosity the subsolution and supersolution of (*) in $\Omega$ with $u\le v$ on $\partial \Omega$. Then we have $u\le v$ in $\Omega$. \par First he proves the theorem under stronger continuity assumptions on $F$. Following an idea of {\it 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. \par 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. \par In the last part of the paper, pointwise estimates for viscosity solutions of (*) (local maximum principle, Harnack inequality, Hölder estimate) are given.
[H.-Ch. Grunau]

MSC 2000:: *35B05 General behavior of solutions of PDE
35J65 (Nonlinear) BVP for (non)linear elliptic equations

Keywords: viscosity solution; fully nonlinear elliptic differential equations; comparison principle; semiconvex; Alexandrov maximum principle; Monge- Ampère equation; Harnack inequality

Citations: Zbl 0708.35019

Cited in: Zbl 1206.35108

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