On global discontinuous solutions of Hamilton-Jacobi equations

Journal article

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamlton-Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H = H(Du), provided the discontinuous initial value function (x) is continuous outside a set Γ of measure zero and satisfies A formula is presented. 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. © 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.

Authors: Chen, G; Su, B

• Journal: Comptes Rendus Mathematique
• Volume: 334
• Issue: 2
• Pages: 113-118
• Publication date: 2002-01-30
• DOI: 10.1016/S1631-073X(02)02228-8
• ISSN: 1631-073X
• Language: English