Uniqueness for a hyperbolic inverse problem with angular control on the coefficients

TL;DR

This paper proves the uniqueness of the potential functions in a hyperbolic inverse problem with angular control, based on boundary measurements of solutions and their derivatives, under a specific geometric and regularity condition.

Contribution

It establishes a uniqueness result for the inverse problem with angular control, extending previous results by incorporating spherical Laplacian conditions.

Findings

Uniqueness of potential functions under angular control

Boundary data determines the potential in an annular region

Condition involving spherical Laplacian is sufficient for uniqueness

Abstract

Suppose $q_i(x)$, $i=1,2$ are smooth functions on $\R^3$ and $U_i(x,t)$ the solutions of the initial value problem {gather*} \pa_t^2 U_i- \Delta U_i - q_i(x) U_i = \delta(x,t), \qquad (x,t) \in \R^3 \times \R U_i(x,t) =0, \qquad \text{for} ~ t<0. {gather*} Pick $R,T$ so that $0 < R < T$ and let $C$ be the vertical cylinder $\{(x,t) \, : |x|=R, ~ R \leq t \leq T \}$. We show that if $(U_1, U_{1r}) = (U_2, U_{2r})$ on $C$ then $q_1 = q_2$ on the annular region $R \leq |x| \leq (R+T)/2$ provided there is a $\gamma>0$, independent of $r$, so that \[\int_{|x|=r} | \Delta_S (q_1 - q_2)|^2 \, dS_x \leq \gamma \int_{|x|=r} |q_1 - q_2|^2 \, dS_x, \qquad \forall r \in [R, (R+T)/2].\] Here $\Delta_S$ is the spherical Laplacian on $|x|=r$.