Uniqueness in an integral geometry problem and an inverse problem for the kinetic equation

Abstract

In this paper, we discuss the uniqueness in an integral geometry problem along the straight lines in a strongly convex domain. Our problem is related with the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation and then the uniqueness theorem is proved using the tools of Fourier analysis.

Keywords:

• Integral geometry problem
• inverse problem
• kinetic equation
• uniqueness

AMS Subject Classifications:

• 35R30
• 53C65

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Most part of the paper has been written during the stay of the second-named author at Department of Mathematical Sciences of The University of Tokyo and the stay was supported by the program “Leading Graduate Course for Frontiers of Mathematical Sciences and Physics”. The third-named author is supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science.