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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Viscosity solutions of Hamilton-Jacobi equations at the boundary


Authors: Michael G. Crandall and Richard Newcomb
Journal: Proc. Amer. Math. Soc. 94 (1985), 283-290
MSC: Primary 35F20; Secondary 70H20
MathSciNet review: 784180
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Abstract: When considering classical solutions of boundary value problems for nonlinear first-order scalar partial differential equations, one knows that there are parts of the boundary of the region under consideration where one cannot specify data and would not expect to require data in order to prove uniqueness. Of course, classical solutions of such problems rarely exist in the large owing to the crossing of characteristics. The theory of a sort of generalized solution--called "viscosity solutions"--for which good existence and uniqueness theorems are valid has been developed over the last few years. In this note we give some results concerning parts of the boundary on which one need not know (prescribe) viscosity solutions to be able to prove comparison (and hence uniqueness) results. In this context, this amounts to identifying boundary points with the property that solutions in the interior which are continuous up to the boundary are also viscosity solutions at the boundary point. Examples indicating the sharpness of the results are given.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0784180-6
PII: S 0002-9939(1985)0784180-6
Keywords: Hamilton-Jacobi equations, viscosity solutions, boundary problems, first order partial differential equations, uniqueness
Article copyright: © Copyright 1985 American Mathematical Society