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

Jensen, R.; Souganidis, P. E.

doi:10.1090/S0002-9947-1987-0879566-1

A regularity result for viscosity solutions of Hamilton-Jacobi equations in one space dimension
HTML articles powered by AMS MathViewer

by R. Jensen and P. E. Souganidis

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

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

PDF | Request permission

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.

References

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
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, DOI 10.1512/iumj.1987.36.36028
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, DOI 10.1017/S0308210500014256
Wendell H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations 5 (1969), 515–530. MR 235269, DOI 10.1016/0022-0396(69)90091-6
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

Uniqueness and stability of the generalized solution of the Cauchy problem for a quasilinear equation

33