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

Yavar Kian (1, 2) , Lauri Oksanen (3)

1 CPT - E8 Dynamique quantique et analyse spectrale
2 CPT - Centre de Physique Théorique - UMR 7332
3 UCL - University College of London [London]

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).

Mots clés: wave equation on manifold; Carleman estimates; partial data; uniqueness; Inverse problems; time-dependent potential

Domaines: Equations aux dérivées partielles [math.AP]

Yavar Kian, Lauri Oksanen. Recovery of time-dependent coefficient on Riemannian manifold for hyperbolic equations. International Mathematics Research Notices, 2019, 2019 (16), pp.5087-5126. ⟨10.1093/imrn/rnx263⟩. ⟨hal-01335372⟩