Abstract
We study the X-ray transform $I$ of symmetric tensor fields on a smooth convex bounded domain $Ω\subset{\mathbb R}^n$. The main result is the stability estimate $\|^{s}f\|_{L^2}≤ C\|If\|_{H^{1/2}}$, where $^{s}f$ is the solenoidal part of the tensor field $f$. The proof is based on a comparison of the Dirichlet integrals for the exterior and interior Dirichlet problems and on a generalization of the Korn inequality to symmetric tensor fields of arbitrary rank.
Highlights
In the Introduction, we use some basic concepts of analysis of symmetric tensor fields without giving definitions
The family of oriented straight lines in Rn. The space S (Rn) is parameterized by points of the manifold
T Sn−1 = {(x, ξ) ∈ Rn × Rn | |ξ| = 1, x, ξ = 0} ⊂ Rn × Rn that is the tangent bundle of the unit sphere Sn−1
Summary
In the Introduction, we use some basic concepts of analysis of symmetric tensor fields (the inner derivative d and divergence δ, the solenoidal part sf of a tensor field f ) without giving definitions. Jan Boman and Vladimir Sharafutdinov where dx is the (n−1)-dimensional Lebesgue measure on ξ⊥ = {x ∈ Rn | x, ξ = 0} This is the standard Fourier transform in the (n−1)-dimensional variable x, where ξ stands as a parameter. In the case of m = 0 (when f is a function), the X-ray transform is the main mathematical tool of Computer Tomography. In the case of m = 1 (when f is a vector field), the operator I is called the Doppler transform and serves as the main mathematical tool of Doppler Tomography. The main difference between scalar and tensor tomography consists of the following: the operator I has a big null-space in the case of m > 0. Is valid with a constant C depending on Ω and m but independent of f
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