Stability estimates in tensor tomography

Abstract: Given a Riemannian metric g on a bounded domain \(Ω ⊂ R^n\) , the geodesic X-ray transform I integrates rank m symmetric tensor fields over maximal geodesics. The problem of inversion of I for m = 2 is the linearization of the boundary rigidity problem. In the case of the Euclidean metric (i.e., when the integration is performed over straight lines), I serves as the main mathematical tool of tomography: Computer Tomography for m = 0, Doppler Tomography for m = 1, Polarization Tomography for m = 2, and some other kinds of tomography of anisotropic media.

Unlike the scalar tomography, the operator I has a big null-space in the case of m > 0. Given If, we can hope to recover the solenoidal part \({s \atop f}\)of a tensor field f only. Under some curvature condition on the metric, Pestov and Sharafutdinov (1988) proved the stability estimate

$$ \|{ ^s \it f }\|_{L^2}^2 \le {C(\| I f \|_{H^1}^2 + {m \| f \| _{H^1} \| \it I \it f \| _{L^2}})}$$ 

The second term on the right-hand side shows that the problem of inversion of I is, perhaps, of a conditionally correct nature: for stably recovering sf from If, we need a priori estimate of \(\|\it f \}_H^1\). The term appears due to the method of the proof, no example is known which demonstrates that the problem is conditionally correct as a matter of fact. The factor m at the second term is distinguished to emphasizes that the problem is well posed in the case of m = 0. In the present work, we consider the case of the Euclidean metric and prove the stronger stability estimate $$ \|{ ^s \it f }\|_{L^2} \le {C \| I  f \|_{H^{1/2}} }$$  for an arbitrary m which demonstrates that the problem is well posed in the Euclidean case. For an arbitrary Riemannian metric g, the question remains open.

In the first lecture, I am going to present all definitions and statements as precisely as I can and to give some motivations; but the first lecture will contain no proof. The second lecture is devoted to the proof of estimate (1). The proof is based on two classical subjects, the Dirichlet principle and Korn’s inequality, that are of some independent interest.