Uniqueness of Solutions to the Helically Reduced Wave Equation with Sommerfeld Boundary Conditions

Abstract

We consider the helical reduction of the wave equation with an arbitrary source on $(n+1)$-dimensional Minkowski space, $n\geq2$. The reduced equation is of mixed elliptic-hyperbolic type on ${\bf R}^n$. We obtain a uniqueness theorem for solutions on a domain consisting of an $n$-dimensional ball $B$ centered on the reduction of the axis of helical symmetry and satisfying ingoing or outgoing Sommerfeld conditions on $\partial B\approx S^{n-1}$. Non-linear generalizations of such boundary value problems (with $n=3$) arise in the intermediate phase of binary inspiral in general relativity.