[Sur des solutions globales discontinues des équations d'Hamilton–Jacobi]
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
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Révisé le :
Publié le :
Gui-Qiang Chen 1 ; Bo Su 2
@article{CRMATH_2002__334_2_113_0, author = {Gui-Qiang Chen and Bo Su}, title = {On global discontinuous solutions of {Hamilton{\textendash}Jacobi} equations}, journal = {Comptes Rendus. Math\'ematique}, pages = {113--118}, publisher = {Elsevier}, volume = {334}, number = {2}, year = {2002}, doi = {10.1016/S1631-073X(02)02228-8}, language = {en}, }
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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