Abstract

Suppose q_i(x), i = 1, 2 are smooth functions on and U_i(x, t) the solutions of the initial value problem , U_i(x, t) = 0 for t < 0. Pick R, T so that 0 < R < T and let C be the vertical cylinder {(x, t) : |x| = R, R ≤ t ≤ T}. We show that if (U ₁, U _1r) = (U ₂, U _2r) on C then q₁ = q₂ on the annular region R ≤ |x| ≤ (R + T)/2 provided there is a γ > 0, independent of r, so that ∫_{|x| = r} | Δ_S(q₁ – q₂)|² dS_x ≤ γ ∫_{|x| = r}|q₁ – q₂|² dS_x for all r ∈ [R, (R + T/2)]. Here Δ_S is the spherical Laplacian on |x| = r.