Global Existence for a Quasilinear Wave Equation Outside of Star-Shaped Domains

We prove global existence of small-amplitude solutions of quasilinear Dirichlet-wave equations outside of star-shaped obstacles in (3+1)-dimensions. We use a variation of the conformal method of Christodoulou. Since the image of the space-time obstacle is not static in the Einstein diamond, our results do not follows directly from local existence theory as did Christodoulouʹs for the nonobstacle case. Instead, we develop weighted estimates that are adapted to the geometry. Using them and the energy-integral method we obtain solutions in the Einstein diamond minus the dime-dependent obstacle, which pull back to solutions in Minkowski space minus and obstacle.