  We consider the strategy of realizing the solution of a Cauchy problem with
radial data as a limit of radial solutions to initial-boundary value problems
posed on the exterior of vanishing balls centered at the origin. The goal is to
gauge the effectiveness of this approach in a simple, concrete setting: the
3-dimensional, linear wave equation $\square_{1+3}U=0$ with radial Cauchy data
$U(0,x)=\Phi(x)=\phi(|x|)$, $U_t(0,x)=\Psi(x)=\psi(|x|)$.
  We are primarily interested in this as a model situation for other, possibly
nonlinear, equations where neither formulae nor abstract existence results are
available for the radial symmetric Cauchy problem. In treating the 3-d wave
equation we therefore insist on robust arguments based on energy methods and
strong convergence. (In particular, this work does not address what can be
established via solution formulae.)
  Our findings for the 3-d wave equation show that while one can obtain
existence of radial Cauchy solutions via exterior solutions, one should not
expect such results to be optimal. The standard existence result for the linear
wave equation guarantees a unique solution in $C([0,T);H^s(\mathbb{R}^3))$
whenever $(\Phi,\Psi)\in H^s\times H^{s-1}(\mathbb{R}^3)$. However, within the
constrained framework outlined above, we obtain strictly lower regularity for
solutions obtained as limits of exterior solutions. We also show that external
Neumann solutions yield better regularity than external Dirichlet solutions.
Specifically, for Cauchy data in $H^2\times H^1(\mathbb{R}^3)$ we obtain
$H^1$-solutions via exterior Neumann solutions, and only $L^2$-solutions via
exterior Dirichlet solutions.

Analysis of PDEs