TY - JOUR

T1 - Convergence of exterior solutions to radial Cauchy solutions for ∂2 t U - c2ΔU = 0

AU - Jenssen, Helge Kristian

AU - Tsikkou, Charis

N1 - Funding Information:
Jenssen was supported by the National Science Foundation under Grant DMS-1311353. Tsikkou was supported by the WVU ADVANCE Sponsorship Program.

PY - 2016/9/30

Y1 - 2016/9/30

N2 - 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.

AB - 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.

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M3 - Article

AN - SCOPUS:84997096187

VL - 2016

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

SN - 1072-6691

M1 - 266

ER -