## Viscosity solutions of Hamilton-Jacobi equations at the boundary

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- by Michael G. Crandall and Richard Newcomb
- Proc. Amer. Math. Soc.
**94**(1985), 283-290 - DOI: https://doi.org/10.1090/S0002-9939-1985-0784180-6
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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.## References

- J.-P. Aubin and I. Ekeland,
- Frank H. Clarke,
*Optimization and nonsmooth analysis*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR**709590** - 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**, DOI 10.1090/S0002-9947-1984-0732102-X - 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**, DOI 10.1090/S0002-9947-1983-0690039-8
—, - 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**, DOI 10.1016/S0304-0208(08)73688-0
R. Jensen, in preparation.
- Pierre-Louis Lions,
*Generalized solutions of Hamilton-Jacobi equations*, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR**667669** - P.-L. Lions,
*Existence results for first-order Hamilton-Jacobi equations*, Ricerche Mat.**32**(1983), no. 1, 3–23 (English, with French summary). MR**740198**
H. M. Soner,

*Nonsmooth analysis*, Working Paper-84-5, Internat. Inst. Appl. Systems Anal., Laxenburg, Austria, 1984.

*On existence and uniqueness of solutions of Hamilton-Jacobi equations*, J. Non. Anal. Th. Meth. Appl. (to appear).

*Optimal control with state space constraint*. I, LCDS Report #84-86, Brown University, 1984.

## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**94**(1985), 283-290 - MSC: Primary 35F20; Secondary 70H20
- DOI: https://doi.org/10.1090/S0002-9939-1985-0784180-6
- MathSciNet review: 784180