Comptes Rendus
no. 2
On global discontinuous solutions of Hamilton–Jacobi equations
[Sur des solutions globales discontinues des équations d'Hamilton–Jacobi]
Gui-Qiang Chen1 ; Bo Su2
1 Department of Mathematics, Northwestern University, Evanston, IL 606037-2730, USA
2 Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1380, USA
Comptes Rendus. Mathématique, Volume 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 y→x,y∈ℝ 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.

Reçu le :
Révisé le :
Publié le :
DOI : 10.1016/S1631-073X(02)02228-8
Gui-Qiang Chen 1 ; Bo Su 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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Gui-Qiang Chen; Bo Su. On global discontinuous solutions of Hamilton–Jacobi equations. Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 113-118. doi : 10.1016/S1631-073X(02)02228-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02228-8/

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