Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations.

*(English)*Zbl 0695.35007The 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.

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 |