Global solutions of a free boundary problem for self-gravitating scalar fields

The weak cosmic censorship hypothesis can be understood as a statement that there exists a global Cauchy evolution of a self-gravitating system outside an event horizon. The resulting Cauchy problem has a free null-like inner boundary. We study a self-gravitating spherically symmetric nonlinear scalar field. We show the global existence of a spacetime with a null inner boundary that initially is located outside the Schwarzschild radius or, more generally, outside an apparent horizon. The global existence of a patch of a spacetime that is exterior to an event horizon is obtained as a limiting case.