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**, 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**, 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**, 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**, 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**, 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.-London, 1982. MR**667669**

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DOI:
https://doi.org/10.1090/S0002-9939-1986-0818466-4

Keywords:
Hamilton-Jacobi equations,
boundary value problems,
viscosity solutions

Article copyright:
© Copyright 1986
American Mathematical Society