摘要
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.
An approach is introduced to construct global discontinuous solutions in L~∞ for HamiltonJacobi 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 discoatinuous 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 to examine 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.
基金
National Science Foundation!DMS-9971793
National Science Foundation!DMS-9708261
关键词
间断解
哈密顿-雅可比方程
黏性解
稳定性
Hamilton-Jacobi equations, Discontinuous solutions, Profit functions, Viscosity solutions, Madman solutions, Stability