We discuss joint work with Markus Keel and Christopher Sogge on global existence of small-amplitude solutions to quasilinear wave equations in 1+3 dimensions, under the condition that the solution vanish on a given compact, smooth, star-shaped obstacle. The key assumption is that the nonlinearities of the equation satisfy the null-condition of Christodoulou and Klainerman. This allows us to exploit the Penrose compactification to transfer the equation to one on the 3-sphere, but with a time-dependent obstacle that is singular at the point corresponding to time-like infinity. Existence is then established through appropriately weighted energy estimates.
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