DISCONTINUOUS SOLUTIONS IN L^∞ FOR HAMILTON-JACOBI EQUATIONS

An approach is introduced to construct global discontinuous solutions in L∞ for Hamilton-Jacobi equations. This approach allows the initial data only in L∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in L∞. The existence of global discontinuous solutions in L∞ is established. These solutions in L∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed toexamine the L∞ stability of our L∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.

DISCONTINUOUS SOLUTIONS IN $L^\infty$ FOR HAMILTON-JACOBI EQUATIONS

An approach is introduced to construct global discontinuous solutions inL ^∞ for Hamilton-Jacobi equations. This approach allows the initial data only inL ^∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions inL ^∞. The existence of global discontinuous solutions inL ^∞ is established. These solutions inL ^∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed to examine theL ^∞ stability of ourL ^∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated. Project supported by the National Science Foundation (DMS-9971793, DMS-9708261).