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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 2 u)=0·(*)

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,vC 0 (Ω ¯)C 0,1 (Ω) be respectively viscosity the subsolution and supersolution of (*) in Ω with uv on Ω. Then we have uv in Ω.

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:
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35J65Nonlinear boundary value problems for linear elliptic equations