A regularity result for viscosity solutions of HamiltonJacobi equations in one space dimension
Authors:
R. Jensen and P. E. Souganidis
Journal:
Trans. Amer. Math. Soc. 301 (1987), 137147
MSC:
Primary 35B65; Secondary 35L60
MathSciNet review:
879566
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Abstract: Viscosity solutions of HamiltonJacobi equations need only to be continuous. Here we prove that, in the special case of a onedimensional 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.
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 [1]
 M. G. Crandall, L. C. Evans, and P.L. Lions, Some properties of viscosity solutions of HamiltonJacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487502. MR 732102 (86a:35031)
 [2]
 M. G. Crandall and P.L. Lions, Viscosity solutions of HamiltonJacobi equations, Trans. Amer. Math. Soc. 277 (1983), 142. MR 690039 (85g:35029)
 [3]
 M. G. Crandall and P. E. Souganidis, Developments in the theory of nonlinear firstorder partial differential equations, Proc. Internat. Sympos. on Differential Equations (Birmingham, Alabama, 1983), I. W. Knowles and R. T. Lewis, Eds., NorthHolland, Amsterdam, 1984. MR 799343 (86j:35033)
 [4]
 P. Cannarsa and H. M. Soner. On the singularities of the viscosity solutions to HamiltonJacobiBellman equations, IMA Preprint Series #201 (to appear). MR 905608 (89m:35044)
 [5]
 C. M. Dafermos, Regularity and large time behaviour of solutions of a conservation law without convexity (to appear). MR 785530 (86j:35107)
 [6]
 W. H. Fleming, The Cauchy problem for a nonlinear firstorder partial differential equation, J. Differential Equations 5 (1969), 515530. MR 0235269 (38:3579)
 [7]
 R. Jensen, in preparation.
 [8]
 P.L. Lions, Generalized solutions of HamiltonJacobi equations, Pitman, Boston, Mass., 1982. MR 667669 (84a:49038)
 [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), 285290.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708795661
PII:
S 00029947(1987)08795661
Keywords:
HamiltonJacobi equations,
viscosity solutions,
regularity
Article copyright:
© Copyright 1987
American Mathematical Society
