Skip to main content Accessibility help
×
×
Home

The Steklov Problem on Differential Forms

  • Mikhail A. Karpukhin (a1)

Abstract

In this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$ -forms on the boundary of a $2p+2$ -dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.

Copyright

References

Hide All
[1] Belishev, M. and Sharafutdinov, V., Dirichlet to Neumann operator on differential forms . Bull. Sci. Math. 132(2008), no. 2, 128145. https://doi.org/10.1016/j.bulsci.2006.11.003.
[2] Fraser, A. and Schoen, R., The first Steklov eigenvalue, conformal geometry, and minimal surfaces . Adv. Math. 226(2011), no. 5, 40114030. https://doi.org/10.1016/j.aim.2010.11.007.
[3] Fraser, A. and Schoen, R., Sharp eigenvalue bounds and minimal surfaces in the ball . Invent. Math. 203(2016), no. 3, 823890. https://doi.org/10.1007/s00222-015-0604-x.
[4] Girouard, A. and Polterovich, I., Shape optimization for low Neumann and Steklov eigenvalues . Math. Methods Appl. Sci. 33(2010), no. 4, 501516. https://doi.org/10.1002/mma.1222.
[5] Girouard, A. and Polterovich, I., Upper bounds for Steklov eigenvalues on surfaces . Electron. Res. Announc. Math. Sci. 19(2012), 7785. https://doi.org/10.3934/era.2012.19.77.
[6] Girouard, A. and Polterovich, I., Spectral geometry of the Steklov problem . J. Spectr. Theory 7(2017), no. 2, 321359. https://doi.org/10.4171/JST/164.
[7] Hersch, J., Payne, L. E., and Schiffer, M. M., Some inequalities for Stekloff eigenvalues . Arch. Rational Mech. Anal. 57(1975), 99114. https://doi.org/10.1007/BF00248412.
[8] Ikeda, A. and Taniguchi, Y., Spectra and eigenforms of the Laplacian on $\mathbb{S}^{n}$ and $P^{n}(\mathbb{C})$ . Osaka J. Math. 15 (1978), no. 3, 515–546.
[9] Joshi, M. S. and Lionheart, W. R. B., An inverse boundary value problem for harmonic differential forms . Asymptot. Anal. 41(2005), no. 2, 93106.
[10] Karpukhin, M., Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds . Electron. Res. Announc. Math. Sci. 24(2017), 100109. https://doi.org/10.3934/era.2017.24.011.
[11] Krupchyk, K., Kurylev, Y., and Lassas, M., Reconstruction of Betti numbers of manifolds for anisotropic Maxwell and Dirac systems . Comm. Anal. Geom. 18(2010), no. 5, 963985. https://doi.org/10.4310/CAG.2010.v18.n5.a4.
[12] Kwong, K.-K., Some sharp Hodge Laplacian and Steklov eigenvalue estimates for differential forms . Calc. Var. Partial Differential Equations 55(2016), no. 2, Art. 38, 14 pp. https://doi.org/10.1007/s00526-016-0977-8.
[13] Raulot, S. and Savo, A., On the first eigenvalue of the Dirichlet-to-Neumann operator on forms . J. Funct. Anal. 262(2012), no. 3, 889914. https://doi.org/10.1016/j.jfa.2011.10.008.
[14] Raulot, S. and Savo, A., On the spectrum of the Dirichlet-to-Neumann operator acting on forms of a Euclidean domain . J. Geom. Phys. 77(2014), 112. https://doi.org/10.1016/j.geomphys.2013.11.002.
[15] Schwarz, G., Hodge decomposition — a method for solving boundary value problems. Lecture Notes in Mathematics, 1607. Springer-Verlag, Berlin, 1995.
[16] Sharafutdinov, V. and Shonkwiler, C., The complete Dirichlet-to-Neumann map for differential forms . J. Geom. Anal. 23(2013), no. 4, 20632080. https://doi.org/10.1007/s12220-012-9320-6.
[17] Shi, Y. and Yu, C., Trace and inverse trace of Steklov eigenvalues . J. Differential Equations 261(2016), no. 3, 20262040. https://doi.org/10.1016/j.jde.2016.04.023.
[18] Shi, Y. and Yu, C., Trace and inverse trace of Steklov eigenvalues II . J. Differential Equations 262(2017), no. 3, 25922607. https://doi.org/10.1016/j.jde.2016.11.018.
[19] Shonkwiler, C., Poincaré duality angles and the Dirichlet-to-Neumann operator . Inverse Problems 29(2013), no. 4. https://doi.org/10.1088/0266-5611/29/4/045007.
[20] Thirring, W., A course in mathematical physics. 2. Second edition. Translated from German by Evans M. Harrell. Springer-Verlag, New York, 1986. https://doi.org/10.1007/978-1-4419-8762-4.
[21] Yang, L. and Yu, C., A higher dimensional generalization of Hersch–Payne–Schiffer inequality for Steklov eigenvalues . J. Funct. Anal. 272(2017), no. 10, 41224130. https://doi.org/10.1016/j.jfa.2017.02.023.
[22] Yang, L. and Yu, C., Estimates for higher Steklov eigenvalues . J. Math. Phys. 58(2017), no. 2. https://doi.org/10.1063/1.4976806.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed