A regularity result for viscosity solutions of Hamilton-Jacobi equations in one space dimension

Authors:
R. Jensen and P. E. Souganidis

Journal:
Trans. Amer. Math. Soc. **301** (1987), 137-147

MSC:
Primary 35B65; Secondary 35L60

MathSciNet review:
879566

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Abstract: Viscosity solutions of Hamilton-Jacobi equations need only to be continuous. Here we prove that, in the special case of a one-dimensional stationary problem, under quite general assumptions, Lipschitz continuous viscosity solutions have right and left derivatives at every point. Moreover, these derivatives have some kind of continuity properties.

**[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 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**[4]**Piermarco Cannarsa and Halil Mete Soner,*On the singularities of the viscosity solutions to Hamilton-Jacobi-Bellman equations*, Indiana Univ. Math. J.**36**(1987), no. 3, 501–524. MR**905608**, 10.1512/iumj.1987.36.36028**[5]**C. M. Dafermos,*Regularity and large time behaviour of solutions of a conservation law without convexity*, Proc. Roy. Soc. Edinburgh Sect. A**99**(1985), no. 3-4, 201–239. MR**785530**, 10.1017/S0308210500014256**[6]**Wendell H. Fleming,*The Cauchy problem for a nonlinear first order partial differential equation*, J. Differential Equations**5**(1969), 515–530. MR**0235269****[7]**R. Jensen, in preparation.**[8]**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****[9]**O. A. Oleinik,*Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation*, Amer. Math. Soc. Transl. (2)**33**(1963), 285-290.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1987-0879566-1

Keywords:
Hamilton-Jacobi equations,
viscosity solutions,
regularity

Article copyright:
© Copyright 1987
American Mathematical Society