zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. (English) Zbl 1142.35041

The authors prove some variants of the comparison principle for viscosity sub- and supersolutions of fully nonlinear elliptic equations F(x,u,Du,D 2 u)=0 in a domain Ω n , extending standard results. Among others, the following case is considered: at any xΩ, F is strictly increasing with respect to u or F is non-totally degenerate, what roughly means that F(x,u,p,M+rI) is a strictly decreasing function of the real parameter r, with I being the identity matrix and M an arbitrary symmetric matrix. A further case refers to operators of the form

F(x,u,p,M)=G(x,u,σ T (x)p,σ T (x)Mσ(x)),

where G is uniformly elliptic and σ is an n×m matrix-valued function satisfying a non-degeneracy condition. A number of more specific equations is considered including Bellman-Isaacs equations, quasilinear subelliptic equations, and equations involving Pucci-type oerators. The results are applied to deduce existence theorems for the Dirichlet problem in the viscosity setting.

MSC:
35J70Degenerate elliptic equations
35J25Second order elliptic equations, boundary value problems
35J65Nonlinear boundary value problems for linear elliptic equations
49L25Viscosity solutions (infinite-dimensional problems)
35H20Subelliptic PDE
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)