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A remark about viscosity solutions of Hamilton-Jacobi equations at the boundary
Author:
P. E. Souganidis
Journal:
Proc. Amer. Math. Soc. 96 (1986), 323-329
MSC:
Primary 35F20; Secondary 35L99
MathSciNet review:
818466
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Abstract: We consider viscosity solutions of first-order partial differential equations of Hamilton-Jacobi type in bounded domains. We give criteria which identify boundary points at which the equation is automatically satisfied in the viscosity sense, if it holds in the interior. These complement some recent results of M. G. Crandall and R. Newcomb [3].
- [1]
M.
G. Crandall, L.
C. Evans, and P.-L.
Lions, Some properties of viscosity solutions
of Hamilton-Jacobi equations, Trans. Amer.
Math. Soc. 282 (1984), no. 2, 487–502. MR 732102
(86a:35031), http://dx.doi.org/10.1090/S0002-9947-1984-0732102-X
- [2]
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
- [3]
Michael
G. Crandall and Richard
Newcomb, Viscosity solutions of Hamilton-Jacobi
equations at the boundary, Proc. Amer. Math.
Soc. 94 (1985), no. 2, 283–290. MR 784180
(86f:35047), http://dx.doi.org/10.1090/S0002-9939-1985-0784180-6
- [4]
Michael
G. Crandall and Panagiotis
E. Souganidis, Developments in the theory of nonlinear first-order
partial differential equations, Differential equations (Birmingham,
Ala., 1983) North-Holland Math. Stud., vol. 92, North-Holland,
Amsterdam, 1984, pp. 131–142. MR 799343
(86j:35033), http://dx.doi.org/10.1016/S0304-0208(08)73688-0
- [5]
Lawrence
C. Evans, Some min-max methods for the Hamilton-Jacobi
equation, Indiana Univ. Math. J. 33 (1984),
no. 1, 31–50. MR 726105
(85b:35009), http://dx.doi.org/10.1512/iumj.1984.33.33002
- [6]
Pierre-Louis
Lions, Generalized solutions of Hamilton-Jacobi equations,
Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing
Program), Boston, Mass., 1982. MR 667669
(84a:49038)
- [1]
- M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487-502. MR 732102 (86a:35031)
- [2]
- 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)
- [3]
- M. G. Crandall and R. Newcomb, Viscosity solutions of Hamilton-Jacobi equations at the boundary, Proc. Amer. Math. Soc. 94 (1985), 283-290. MR 784180 (86f:35047)
- [4]
- M. G. Crandall and P. E. Souganidis, Developments in the theory of nonlinear first-order partial differential equations (Proc. Internat. Sympos. on Differential Equations, Birmingham, Alabama, 1983, I. W. Knowles and R. T. Lewis, eds.), North-Holland, Amsterdam, 1984. MR 799343 (86j:35033)
- [5]
- L. C. Evans, Some max-min methods for the Hamilton-Jacobi equation, Indiana Univ. Math. J. 33 (1984), 31-50. MR 726105 (85b:35009)
- [6]
- P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations, Pitman, Boston, Mass., 1982. MR 667669 (84a:49038)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1986-0818466-4
PII:
S 0002-9939(1986)0818466-4
Keywords:
Hamilton-Jacobi equations,
boundary value problems,
viscosity solutions
Article copyright:
© Copyright 1986 American Mathematical Society
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