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Stability in the Inverse Steklov Problem on Warped Product Riemannian Manifolds

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Abstract

In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in \({\mathbb {R}}^d\). We first prove that the Steklov spectrum determines uniquely the warping function of the metric. We show in fact that the approximate knowledge (in a given precise sense) of the Steklov spectrum is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, we provide stability estimates of log-type on the warping function from the Steklov spectrum. The key element of these stability results relies on a formula that, roughly speaking, connects the inverse data (the Steklov spectrum) to the Laplace transform of the difference of the two warping factors.

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Notes

  1. Observe that the condition \(u \in H^1(M)\) can be written in terms of \(v_k\) (after separation of variables) as

    $$\begin{aligned} \sum _{k=0}^\infty \int _0^{+\infty } \left[ f^4(x)|v_k|^2 + \mu _k |v_k|^2 + f^{2d-4}(x) |(f^{2-d} v_k)'|^2 \right] dx < \infty . \end{aligned}$$

    We refer to [7] for the details of a similar calculation. Hence, for \(k \ge 1\), we see that \(v_k \in L^2\) from the second term under the integral. For \(k=0\), we can use the third term under the integral to prove that \(v_0 \in L^2\).

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Acknowledgements

We thank the two anonymous referees for their careful reading and constructive suggestions. Thierry Daudé’s research is supported by the JCJC French National Research Projects Horizons, No. ANR-16-CE40-0012-01, Niky Kamran’s research is supported by NSERC grant RGPIN 105490-2018 and Francois Nicoleau’s research is supported by the French National Research Project GDR Dynqua.

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Authors and Affiliations

  1. Département de Mathématiques, UMR CNRS 8088, Université de Cergy-Pontoise, 95302, Cergy-Pontoise, France

    Thierry Daudé

  2. Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, Canada

    Niky Kamran

  3. Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 2 Rue de la Houssinière, BP 92208, 44322, Nantes Cedex 03, France

    François Nicoleau

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  1. Thierry Daudé
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  2. Niky Kamran
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  3. François Nicoleau
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Correspondence to François Nicoleau.

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Thierry Daudé: Research supported by the French National Research Projects AARG, No. ANR-12-BS01-012-01, and Iproblems, No. ANR-13-JS01-0006.

Niky Kamran: Research supported by NSERC grant RGPIN 105490-2011.

François Nicoleau: Research supported by the French GDR Dynqua.

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Daudé, T., Kamran, N. & Nicoleau, F. Stability in the Inverse Steklov Problem on Warped Product Riemannian Manifolds. J Geom Anal 31, 1821–1854 (2021). https://doi.org/10.1007/s12220-019-00326-9

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