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A class of Hamilton-Jacobi equations arising in generalized timeoptimal control problems and differential games is considered. The natural global boundary value problem for these equations has a singular boundary condition on a free boundary. The uniqueness of the continuous solution and of the free boundary is proved in the framework of viscosity solutions. A local uniqueness theorem is also given, as well as some existence results and several applications to control and game theory. Indoi:10.1090/s0002-9947-1991-0991958-x fatcat:tsyvvwh6x5b6rhe77f3fa4vnj4