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Spectral invariants of the perturbed polyharmonic Steklov problem

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Abstract

For a given bounded domain \(\Omega \) with smooth boundary in a smooth Riemannian manifold \(({\mathcal {M}},g)\), we establish a procedure to calculate all the coefficients of the asymptotic expansion for the heat kernel trace of the Dirichlet-to-Neumann map \(\Lambda _m\) (\(m\ge 1\)) associated with perturbed polyharmonic operator as \(t\rightarrow 0^+\). We also explicitly calculate the first four coefficients of this asymptotic expansion. These coefficients (i.e., heat invariants) provide precise information for the area and curvatures of the boundary \(\partial \Omega \) in terms of the spectrum of the perturbed polyharmonic Steklov problem. In particular, when \(m=1\) and \(q\equiv 0\) our work recovers the previous corresponding results in Liu (J Differ Equ 259, 2499–2545, 2015) and Polterovich and Sher (J Geom Anal 25: 924–950, 2015).

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Acknowledgements

I wish to express my sincere gratitude to Professor L. Nirenberg and Professor Fang-Hua Lin for their great support and help. I would like to thank the Editor and an anonymous referee for many valuable comments and suggestions. This research was supported by NNSF of China (11671033/A010802) and NNSF of China (11171023/A010801).

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  1. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, The People’s Republic of China

    Genqian Liu

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Communicated by F.-H. Lin.

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Liu, G. Spectral invariants of the perturbed polyharmonic Steklov problem. Calc. Var. 61, 125 (2022). https://doi.org/10.1007/s00526-022-02183-x

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