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)
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. (English) Zbl 0708.35031

This is a long and technical paper about viscosity solutions for fully nonlinear elliptic equations F(x,u,u,D 2 u)=0 in Ω under various boundary conditions. The common strategy to tackle these equations is the observation of the first author [Duke Math. J. 55, 362- 384 (1987) and Commun. Pure Appl. Math. 42, No., 15-45 (1989; Zbl 0645.35025)] that (unique) existence is implied by a Perron-process, if viscosity sub- and supersolutions are known and a kind of maximum principle can be proved. It reads as follows: Whenever u (resp. v) is an usc (resp. lsc) bounded viscosity sub- (resp. super-) solution, then

u-vsup xΩ {u * (x)-v * (x)} + inΩ

sup yΩ,yx supu(y) and v * =-(-v) * ·

Hence the problem remains in (and most of the paper is devoted to) veryfying this under various structure conditions on F, including Isaac- Bellman equations and also Monge-Ampère equations. See also R. Jensen [Arch. Ration. Mech. Anal. 101, No.1, 1-27 (1988)]. The paper closes with some remarks to the regularity of solutions. For C 1,α -estimates, see also N. S. Trudinger [Proc. R. Soc. Edinb. Sect. A 108, No.1/2, 57-65 (1988; Zbl 0653.35026)].

Reviewer: M.Wiegner
35J65Nonlinear boundary value problems for linear elliptic equations
35B50Maximum principles (PDE)
35B65Smoothness and regularity of solutions of PDE
35J70Degenerate elliptic equations
35D10Regularity of generalized solutions of PDE (MSC2000)