Consider the Cauchy problem for the 3-D linear wave equation ∂2t U - c2ΔU = 0 with radial initial data U(0, x) = Φ(x) = φ(|x|), Ut(0, x) = Ψ(x) = ψ (|x|). A standard result states that U belongs to C([0, T];Hs(ℝ3)) whenever (Φ, Ψ) ∈ Hs × Hs-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 Bε about the origin. We note that, as the solutions we compare are defined on different domains, the answer is not an immediate consequence of Hs 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 Bε 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 \ Bε 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.
|Original language||English (US)|
|Journal||Electronic Journal of Differential Equations|
|State||Published - Sep 30 2016|
All Science Journal Classification (ASJC) codes