zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

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