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

Abstract

We consider the inverse problem of determining a time-dependent potential $q$, appearing in the wave equation $\partial_t^2u-Δu+q(t,x)u=0$ in $Q=(0,T)\timesΩ$ with $Ω$ a $C^2$ bounded domain of $\mathbb R^n$, $n\geq2$, from partial observations of the solutions on $\partial Q$. We prove global unique determination of a coefficient $q\in L^\infty(Q)$ from these observations.

The previous version of the paper arXiv:1406.5734 is now decomposed into two different papers with a new uniqueness result (that improves the previous one with weaker conditions and extension to the case of dimension 2) stated in the present paper and a new stability estimate (with different approach than the pervious version and extension to the case of dimension 2) stated in arXiv:1406.5734