Communications in Analysis and Geometry

Volume 31 (2023)

Number 7

Determination of a compact Finsler manifold from its boundary distance map and an inverse problem in elasticity

Pages: 1693 – 1747

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n7.a4

Authors

Maarten V. de Hoop (Simons Chair in Computational and Applied Mathematics and Earth Science, Rice University, Houston, Texas, U.S.A.)

Joonas Ilmavirta (Unit of Computing Sciences, Tampere University, Tampere, Finland; and Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Matti Lassas (Department of Mathematics and Statistics, University of Helsinki, Finland)

Teemu Saksala (Department of Computational and Applied Mathematics, Rice University, Houston, Texas, U.S.A.; and Department of Mathematics, North Carolina State University, Raleigh, N.C., U.S.A.)

Abstract

We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differentiable structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points.

We recall how the fastest $qP$-polarized waves in anisotropic elastic medium are a given as solutions of the second order hyperbolic pseudo-differential equation $(\frac{\partial^2}{\partial t^2} \lambda^1 (x,D)) u(t,x) = h(t, x)$ on $\mathbb{R}^{1+3}$, where $\sqrt{\lambda^1}$ is the Legendre transform of a fiberwise real analytic Finsler function $F$ on $\mathbb{R}^3$. If $M \subset \mathbb{R}^3$ is a $F$-convex smooth bounded domain we say that a travel time of $u$ to $z \in \partial M$ is the first time $t \gt 0$ when the wavefront set of $u$ arrives in $(t, z)$. The aforementioned geometric result can then be utilized to determine the isometry class of $(\overline{M}, F)$ if we have measured a large amount of travel times of $qP$-polarized waves, issued from a dense set of unknown interior point sources on $M$.

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MVdH was supported by the Simons Foundation under the MATH + X program, the National Science Foundation under grant DMS-1815143, and by the members of the Geo-Mathematical Imaging Group at Rice University.

JI was supported by the Academy of Finland (decisions 295853, 332890, and 336254). Much of the work was completed during JI’s visits to Rice University, and he is grateful for hospitality and support.

ML was supported by Academy of Finland (decisions 284715 and 303754).

TS was supported by the Simons Foundation under the MATH + X program.

Received 22 December 2020

Accepted 16 July 2021

Published 10 August 2024