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 sys tem outside an event horizon. The resulting Cauchy problem has a free null-like inner boundary. We study a self-gravitating spherically symm etric nonlinear scalar field. We show the global existence of a spacet ime with a null inner boundary that initially is located outside the S chwarzschild radius or, more generally, outside an apparent horizon. T he global existence of a patch of a spacetime that is exterior to an e vent horizon is obtained as a limiting case.