On global discontinuous solutions of Hamilton–Jacobi equations

Chen, Gui-Qiang; Su, Bo

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

$$\varphi \left(\mathrm{x}\right)⩾{\varphi }_{**}\left(\mathrm{x}\right):=\underset{\mathrm{y}\to \mathrm{x},\mathrm{y}\in {ℝ}^d⧹\Gamma }{\lim \inf }\varphi \left(\mathrm{y}\right).$$

(∗)

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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