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

Abstract : Given (M, g), a compact connected Riemannian manifold of dimension d 2, with boundary ∂M , we study the inverse boundary value problem of determining a time-dependent potential q, appearing in the wave equation ∂ 2 t u−∆gu+q(t, x)u = 0 in M = (0, T)×M with T > 0. Under suitable geometric assumptions we prove global unique determination of q ∈ L ∞ (M) given the Cauchy data set on the whole boundary ∂M , or on certain subsets of ∂M. Our problem can be seen as an analogue of the Calderón problem on the Lorentzian manifold (M , dt 2 − g).
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Yavar Kian, Lauri Oksanen. Recovery of time-dependent coefficient on Riemannian manifold for hyperbolic equations. International Mathematical Research Notices, Oxford University Press, 2017, pp.rnx263. ⟨10.1093/imrn/rnx263⟩. ⟨hal-01335372⟩

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