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A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations
Authors:
R. Jensen, P.-L. Lions and P. E. Souganidis
Journal:
Proc. Amer. Math. Soc. 102 (1988), 975-978
MSC:
Primary 35B25; Secondary 35J60
MathSciNet review:
934877
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Abstract: In this note we extend some recent results of R. Jensen concerning the uniqueness of viscosity solutions of scalar, second order, fully nonlinear, elliptic, possibly degenerate, partial differential equations.
- [1]
A. D. Alexandrov, Majorization of solutions of second-order linear equations, Amer. Math. Soc. Transl. 68 (1968), 120-143.
- [2]
Jean-Michel
Bony, Principe du maximum dans les espaces de Sobolev, C. R.
Acad. Sci. Paris Sér. A-B 265 (1967),
A333–A336 (French). MR 0223711
(36 #6759)
- [3]
Michael
G. Crandall, Hitoshi
Ishii, and Pierre-Louis
Lions, Uniqueness of viscosity solutions of Hamilton-Jacobi
equations revisited, J. Math. Soc. Japan 39 (1987),
no. 4, 581–596. MR 905626
(88k:35038), http://dx.doi.org/10.2969/jmsj/03940581
- [4]
Michael
G. Crandall and Pierre-Louis
Lions, Viscosity solutions of Hamilton-Jacobi
equations, Trans. Amer. Math. Soc.
277 (1983), no. 1,
1–42. MR
690039 (85g:35029), http://dx.doi.org/10.1090/S0002-9947-1983-0690039-8
- [5]
Wendell
H. Fleming and Raymond
W. Rishel, Deterministic and stochastic optimal control,
Springer-Verlag, Berlin, 1975. Applications of Mathematics, No. 1. MR 0454768
(56 #13016)
- [6]
W.
H. Fleming and P.
E. Souganidis, Two-player, zero-sum stochastic differential
games, Analyse mathématique et applications, Gauthier-Villars,
Montrouge, 1988, pp. 151–164. MR 956957
(89g:93090)
- [7]
Robert
Jensen, The maximum principle for viscosity solutions of fully
nonlinear second order partial differential equations, Arch. Rational
Mech. Anal. 101 (1988), no. 1, 1–27. MR 920674
(89a:35038), http://dx.doi.org/10.1007/BF00281780
- [8]
Robert
Jensen and Pierre-Louis
Lions, Some asymptotic problems in fully nonlinear elliptic
equations and stochastic control, Ann. Scuola Norm. Sup. Pisa Cl. Sci.
(4) 11 (1984), no. 1, 129–176. MR 752583
(85m:49054)
- [9]
J.-M.
Lasry and P.-L.
Lions, A remark on regularization in Hilbert spaces, Israel J.
Math. 55 (1986), no. 3, 257–266. MR 876394
(88b:41020), http://dx.doi.org/10.1007/BF02765025
- [10]
P.-L.
Lions, Optimal control of diffusion processes and
Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and
uniqueness, Comm. Partial Differential Equations 8
(1983), no. 11, 1229–1276 (English, with French summary). MR 709162
(85i:49043b), http://dx.doi.org/10.1080/03605308308820301
- [11]
P.-L.
Lions, Fully nonlinear elliptic equations and applications,
Nonlinear analysis, function spaces and applications, Vol. 2 (Písek,
1982), Teubner-Texte zur Math., vol. 49, Teubner, Leipzig, 1982,
pp. 126–149. MR 684999
(84b:35045)
- [12]
P.-L.
Lions, A remark on Bony maximum
principle, Proc. Amer. Math. Soc.
88 (1983), no. 3,
503–508. MR
699422 (84m:35044), http://dx.doi.org/10.1090/S0002-9939-1983-0699422-3
- [1]
- A. D. Alexandrov, Majorization of solutions of second-order linear equations, Amer. Math. Soc. Transl. 68 (1968), 120-143.
- [2]
- J. M. Bony, Principle du maximum dans les espaces de Sobolev, C.R. Acad. Sci. Paris 265 (1967), 333-336. MR 0223711 (36:6759)
- [3]
- M. G. Crandall, H. Ishii and P.-L. Lions, Uniqueness of viscosity solutions revisited, J. Math. Soc. Japan 39 (1987), 581-598. MR 905626 (88k:35038)
- [4]
- M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1-42. MR 690039 (85g:35029)
- [5]
- W. H. Fleming and R. W. Richel, Deterministic and stochastic optimal control, Springer-Verlag, Berlin and New York, 1975. MR 0454768 (56:13016)
- [6]
- W. H. Fleming and P. E. Souganidis, Value functions of two players, zero-sum stochastic differential games (to appear). MR 956957 (89g:93090)
- [7]
- R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equation, Arch. Rational Mech. Anal. (to appear). MR 920674 (89a:35038)
- [8]
- R. Jensen and P.-L. Lions, Some asymptotic problems in fully nonlinear elliptic equations and stochastic control, Ann. Sci. École Norm. Sup. Pisa 11 (1983), 129-176. MR 752583 (85m:49054)
- [9]
- J. M. Lasry and P.-L. Lions, A remark on regularization on Hilbert spaces, Israel J. Math. (to appear). MR 876394 (88b:41020)
- [10]
- P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II, Comm. Partial Differential Equations 8 (1983), 1229-1276. MR 709162 (85i:49043b)
- [11]
- -, Fully nonlinear elliptic equations and applications, Nonlinear Analysis, Function Spaces and Applications, Teubner, Leipzig, 1982. MR 684999 (84b:35045)
- [12]
- -, A remark on the Bony maximum principle, Proc. Amer. Math. Soc. 88 (1983), 503-508. MR 699422 (84m:35044)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1988-0934877-2
PII:
S 0002-9939(1988)0934877-2
Keywords:
Viscosity solutions,
second order nonlinear partial differential equations,
degenerate elliptic,
uniqueness
Article copyright:
© Copyright 1988 American Mathematical Society
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