The John equation for tensor tomography in three-dimensions

Nikolai Nadirashvili (1) , Serge G Vlădut ̧ (1) , Vladimir Sharafutdinov (2)

1 I2M - Institut de Mathématiques de Marseille
2 Sobolev Institute of Mathematics

John proved that a function $j$ on the manifold of lines in $R^3$ belongs to the range of the x-ray transform if and only if $j$ satisfies some second order differential equation and obeys some smoothness and decay conditions. We generalize the John equation to the case of the x-ray transform on arbitrary rank symmetric tensor fields: a function j on the manifold of lines in$R^3$ belongs to the range of the x-ray transform on rank m symmetric tensor fields if and only if $j$ satisfies some differential equation of order 2(m + 1) and obeys some smoothness and decay conditions.

Domaines: Equations aux dérivées partielles [math.AP]

Nikolai Nadirashvili, Serge G Vlădut ̧, Vladimir Sharafutdinov. The John equation for tensor tomography in three-dimensions. Inverse Problems, 2016, 32 (2016) pp.105013. ⟨hal-01485393⟩