On global discontinuous solutions of Hamilton–Jacobi equations
[Sur des solutions globales discontinues des équations d'Hamilton–Jacobi]
Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 113-118.

On établit l'unicité des solutions de viscosité semicontinues classiques du problème de Cauchy des équations d'Hamilton–Jacobi possèdant des Hamiltonien H=H(Du) convexe et Lipschitz continue globale, si la fonction initiale discontinue ϕ(x) est continue à l'extérieur de l'ensemble Γ de mesure zéro et satisfait (*). On montre la régularité des solutions discontinues des équations d'Hamilton–Jacobi possédant des Hamiltoniens localement strictement convexes : les solutions discontinues possédant les données initiales continues presque partout et satisfaisant (*) deviennent Lipschitz continues après un temps fini. On prouve la L1-accessibilité des données initiales et un principe de comparaison. On clarifie aussi l'équivalence des solutions de viscosité semicontinues, des solutions bi-latérales, des L-solutions, des solutions minimax, et des L-solutions.

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous initial value function ϕ(x) is continuous outside a set Γ of measure zero and satisfies

ϕ(x)ϕ ** (x):= lim inf yx,y d Γϕ(y).(∗)
We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L1-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L-solutions is clarified.

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Révisé le :
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DOI : 10.1016/S1631-073X(02)02228-8
Chen, Gui-Qiang 1 ; Su, Bo 2

1 Department of Mathematics, Northwestern University, Evanston, IL 606037-2730, USA
2 Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1380, USA
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Chen, Gui-Qiang; Su, Bo. On global discontinuous solutions of Hamilton–Jacobi equations. Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 113-118. doi : 10.1016/S1631-073X(02)02228-8. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02228-8/

[1] Barles, G.; Perthame, B. Discontinuous solutions of deterministic optimal stopping problem, Math. Model. Numer. Anal., Volume 2 (1987), pp. 557-579

[2] Barron, E.N.; Jensen, R. Semicontinuous viscosity solutions of Hamilton–Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, Volume 15 (1990), pp. 1713-1742

[3] Chen, G.-Q.; Su, B. Discontinuous solutions in L for Hamilton–Jacobi equations, Chinese Ann. Math., Volume 2 (2000), pp. 165-186

[4] Chen G.-Q., Su B., On discontinuous solutions of Hamilton–Jacobi equations, Preprint, June 2001

[5] Crandall, M.; Lions, P.-L. Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc., Volume 277 (1983), pp. 1-42

[6] Crandall, M.; Ishii, H.; Lions, P.-L. A user's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc., Volume 27 (1992), pp. 1-67

[7] Demengel, F.; Serre, D. Nonvanishing singular parts of measure valued solutions for scalar hyperbolic equations, Comm. Partial Differential Equations, Volume 16 (1991), pp. 221-254

[8] Giga Y., Sato M.H., A level set approach to semicontinuous solutions for Cauchy problems, Preprint, 2001

[9] Glimm, J.; Kranzer, H.C.; Tan, D.; Tangerman, F.M. Wave fronts for Hamilton–Jacobi equations: the general theory for Riemann solutions in R n , Comm. Math. Phys., Volume 187 (1997), pp. 647-677

[10] Ishii, H. Uniqueness of unbounded viscosity solution of Hamilton–Jacobi equations, Indiana Univ. Math. J., Volume 33 (1984), pp. 721-748

[11] Ishii, H. Perron's method for Hamilton–Jacobi equations, Duke Math. J., Volume 55 (1987), pp. 368-384

[12] Kruzhkov, S.N. Generalized solutions of nonlinear equations of the first order with several independent variables, II, Mat. Sb. (N.S.), Volume 114 (1967), pp. 108-134 (in Russian)

[13] Lions, P.-L. Generalized Solutions of Hamilton–Jacobi Equations, Research Notes in Math., 69, Pitman, Boston, 1982

[14] Liu, T.-P.; Pierre, M. Source solutions and asymptotic behavior in conservation laws, J. Differential Equations, Volume 51 (1984), pp. 419-441

[15] Subbotin, A.I. Generalized Solutions of First Order PDEs, Birkhäuser, Boston, 1995

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