Discontinuous solutions in L(infinity) of Hamilton-Jacobi equations
We introduce a general approach to construct global discontinuous solutions of Hamilton-Jacobi equations. Our approach allows the Cauchy data only in L∞ and applies to the equations with non-convex Hamiltonians. The profit functions are introduced to formulate our new notion of discontinuous solutions. The existence of global discontinuous solutions in L∞ is established. Our 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 uniqueness and stability of discontinuous solutions.