A uniqueness result in an inverse hyperbolic problem with analyticity

Abstract

We prove the uniqueness for the inverse problem of determining a coefficient $q(x)$ in $\partial _t^2 u(x,t) = \uDelta u(x,t) - q(x)u(x,t)$ for $x \in R^n$ and $t > 0$, from observations of $u\vert_{\Gamma\times(0,T)}$ and the normal derivative $\frac{\partial u}{\partial \nu}\vert_{\Gamma\times(0,T)}$ where $\Gamma$ is an arbitrary $C^{\infty}$-hypersurface. Our main result asserts the uniqueness of $q$ over $R^n$ provided that $T > 0$ is sufficiently large and $q$ is analytic near $\Gamma$ and outside a ball. The proof depends on Fritz John's global Holmgren theorem and the uniqueness by a Carleman estimate.