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)
The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. (English) Zbl 0708.35019
Der Autor betrachtet voll nichtlineare elliptische Differentialgleichungen F(D 2 u,Du,u)=0 zweiter Ordnung. Er weist nach, daß Viskositätslösungen in W 1, (Ω)C(Ω ¯) eindeutig sind sofern a) F entartet elliptisch und monoton fallend in u ist, oder b) F gleichmäßig elliptisch und monoton nicht wachsend in u ist. Beim Beweis werden Regularisierungen von u benutzt, welche Viskositäts- Ober- und Unterlösungen in Viskositäts- Ober- und Unterlösungen überführen. Frühere Arbeiten von P. L. Lions hatten Konvexität oder Konkavität von F und lineares Wachstum in (D 2 u,Du,u) vorausgesetzt, allerdings auch Abhängigkeit von x zugelassen.
Reviewer: B.Kawohl

MSC:
35B50Maximum principles (PDE)
35J60Nonlinear elliptic equations
References:
[1]J. M. Bony, Principe du maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris 265 (1967), 333-336.
[2]M. G. Crandall, L. C. Evans, & P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984), 487-502. · doi:10.1090/S0002-9947-1984-0732102-X
[3]M. G. Crandall & P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1-42. · doi:10.1090/S0002-9947-1983-0690039-8
[4]L. C. Evans, A convergence theorem for solutions of nonlinear second order elliptic equations. Indiana Univ. Math. J. 27 (1978), 875-887. · Zbl 0408.35037 · doi:10.1512/iumj.1978.27.27059
[5]L. C. Evans, On solving certain nonlinear partial differential equations by accretive operator methods. Israel J. Math. 36 (1980), 225-247. · Zbl 0454.35038 · doi:10.1007/BF02762047
[6]L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations. Comm. Pure Appl. Math. 25 (1982), 333-363. · Zbl 0477.35016 · doi:10.1002/cpa.3160350303
[7]P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1. Dynamic Programming Principle and applications. Comm. Partial Differential Equations 8 (1983), 1101-1134. · Zbl 0716.49022 · doi:10.1080/03605308308820297
[8]P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 2. Viscosity solutions and uniqueness. Comm. Partial Differential Equations 8 (1983), 1229-1276. · Zbl 0716.49023 · doi:10.1080/03605308308820301
[9]P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 3. Regularity of the optimal cost function. Collège de France Seminar, Vol. V, Pitman, London, 1983.
[10]C. Pucci, Limitazioni per soluzioni di equazioni ellitiche. Ann. Mat. Pura Appl. 74 (1966), 15-30. · Zbl 0144.35801 · doi:10.1007/BF02416445
[11]E. M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, New Jersey, 1970.