Uniqueness Theorems for Linear Wave Equations (Paperback)

We prove global uniqueness theorems for linear wave equations in the Minkowski and Schwarzschild spacetimes. In the Minkowski spacetime we show that in the class of solutions with bounded energy and L 2 norm, we have uniqueness across planes for a wide class of equations. The proof uses a new degenerate Carleman estimate plus the energy bounds to show that certain error terms are small. The proof is more robust than previous results in this direction and works for a more general class of metrics beyond the flat case. In Schwarzschild, we prove two results. First, we show that uniqueness holds for smooth linear wave equations across the union of the future event horizon and a past null cone outside the event horizon. This follows from showing the existence of a novel family of pseudoconvex surfaces, and then applying usual Carleman estimates. The final result shows that for the Klein-Gordon equation in Schwarzschild to prove uniqueness globally for finite energy solutions we need only information on the future null cone (and not a bifurcate null hypersurface). This is shown by proving an integral inequality for energy estimates related to the vector field Z = u2du + v 2dv.