Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations

Abstract

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d\ge 2$, with boundary $\partial M$, we study the inverse boundary value problem of determining a time-dependent potential $q$, appearing in the wave equation ${\partial }_t^{2}u-{\Delta }_{g}u+q(t,x)u=0$ in $\bar{M}=(0,T)\times M$ with $T>0$. Under suitable geometric assumptions we prove global unique determination of $q\in L^{\infty }(\bar{M})$ given the Cauchy data set on the whole boundary $\partial \bar{M}$, or on certain subsets of $\partial \bar{M}$. Our problem can be seen as an analogue of the Calderón problem on the Lorentzian manifold $(\bar{M},d{t}^2-g)$.

References in corpus (1)

• Unique determination of a time-dependent potential for wave equations from partial data

Cited by in corpus (2)

• Application of the boundary control method to partial data Borg-Levinson inverse spectral problem
• Determination of singular time-dependent coefficients for wave equations from full and partial data