Convergence of exterior solutions to radial Cauchy solutions for ∂² _t U - c²ΔU = 0

Consider the Cauchy problem for the 3-D linear wave equation ∂²_t U - c²ΔU = 0 with radial initial data U(0, x) = Φ(x) = φ(|x|), U_t(0, x) = Ψ(x) = ψ (|x|). A standard result states that U belongs to C([0, T];H^s(ℝ³)) whenever (Φ, Ψ) ∈ H^s × H^s-1(ℝ³). In this article we are interested in the question of how U can be realized as a limit of solutions to initial-boundary value problems on the exterior of vanishing balls B_ε about the origin. We note that, as the solutions we compare are defined on different domains, the answer is not an immediate consequence of H^s well-posedness for the wave equation. We show how explicit solution formulae yield convergence and optimal reg- ularity for the Cauchy solution via exterior solutions, when the latter are extended continuously as constants on B_ε at each time. We establish that for s = 2 the solution U can be realized as an H²-limit (uniformly in time) of exterior solutions on ℝ³ \ B_ε satisfying vanishing Neumann conditions along |x| = ε, as ε ↓ 0. Similarly for s = 1: U is then an H¹-limit of exterior solutions satisfying vanishing Dirichlet conditions along |x| = ε.