by Joonas Ilmavirta, Mikko Salo
Abstract:
We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of the obstacle. Our proof is based on a Pestov identity with boundary terms, and it involves Jacobi fields on broken rays. We also discuss applications of the broken ray transform.
Reference:
Broken ray transform on a Riemann surface with a convex obstacle (Joonas Ilmavirta, Mikko Salo), Communications in Analysis and Geometry, volume 24, number 2, pp. 379–408, 2016.
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We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of the obstacle. Our proof is based on a Pestov identity with boundary terms, and it involves Jacobi fields on broken rays. We also discuss applications of the broken ray transform.
[arXiv]
Bibtex Entry:
@article{brt-pde-1obst,
author = {Joonas Ilmavirta and Mikko Salo},
title = {{Broken ray transform on a Riemann surface with a convex obstacle}},
journal = {Communications in Analysis and Geometry},
month = jun,
year = {2016},
volume = {24},
number = {2},
pages = {379--408},
arxiv = {1403.5131},
doi = {10.4310/CAG.2016.v24.n2.a6},
url={http://users.jyu.fi/~jojapeil/pub/brt-pde-1obst.pdf},
gsid = {8734444451560463526},
abstract = {We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of the obstacle. Our proof is based on a Pestov identity with boundary terms, and it involves Jacobi fields on broken rays. We also discuss applications of the broken ray transform.}
}