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Spectrum of the Laplacian with weights

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Abstract

Given a compact Riemannian manifold (M, g) and two positive functions \(\rho \) and \(\sigma \), we are interested in the eigenvalues of the Dirichlet energy functional weighted by \(\sigma \), with respect to the \(L^2\) inner product weighted by \(\rho \). Under some regularity conditions on \(\rho \) and \(\sigma \), these eigenvalues are those of the operator \( -\rho ^{-1} \text{ div }(\sigma \nabla u) \) with Neumann conditions on the boundary if \(\partial M\ne \emptyset \). We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.

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References

  1. Anné, C.: Spectre du laplacien et écrasement d’anses. Ann. Sci. École Norm. Sup. (4) 20(2), 271–280 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Astala, K., Päivärinta, L.: Calderón’s inverse conductivity problem in the plane. Ann. Math. (2) 163(1), 265–299 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bérard, P.H.: Spectral Geometry: Direct and Inverse Problems, volume of 1207 Lecture Notes in Mathematics. Springer, Berlin (1986)

    Google Scholar 

  4. Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété riemannienne, volume of 194 Lecture Notes in Mathematics. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  5. Buser, P.: On Cheeger’s inequality \(\lambda_{1}\ge h^{2}/4\). In: Geometry of the Laplace operator (Proceedings of Symposium on Pure Mathematics, University of Hawaii, Honolulu, Hawaii, 1979), Proceedings of Symposium on Pure Mathematics, vol. XXXVI, pp. 29–77. American Mathematical Society, Providence (1980)

  6. Buser, P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15(2), 213–230 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chavel, I.: Eigenvalues in Riemannian Geometry, volume 115 of Pure and Applied Mathematics. Academic Press, Orlando (1984). (Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk)

    Google Scholar 

  8. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton University Press, Princeton (1970)

  9. Colbois, B., El Soufi, A.: Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ‘conformal spectrum’. Ann. Glob. Anal. Geom. 24(4), 337–349 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Colbois, B., El Soufi, A.: Eigenvalues of the Laplacian acting on \(p\)-forms and metric conformal deformations. Proc. Am. Math. Soc. 134(3), 715–721 (2006). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  11. Colbois, B., El Soufi, A., Girouard, A.: Isoperimetric control of the spectrum of a compact hypersurface. J. Reine Angew. Math. 683, 49–65 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Colbois, B., El Soufi, A., Savo, A.: Eigenvalues of the Laplacian on a manifold with density. Commun. Anal. Geom. 23(3), 639–670 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colbois, B., Maerten, D.: Eigenvalues estimate for the Neumann problem of a bounded domain. J. Geom. Anal. 18(4), 1022–1032 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Colin de Verdière, Y.: Sur la multiplicité de la première valeur propre non nulle du laplacien. Comment. Math. Helv. 61(2), 254–270 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. El Soufi, A.: Applications harmoniques, immersions minimales et transformations conformes de la sphère. Compos. Math. 85(3), 281–298 (1993)

    MATH  Google Scholar 

  16. El Soufi, A., Harrell II, E.M., Ilias, S., Stubbe, J.: On sums of eigenvalues of elliptic operators on manifolds. J. Spectr. Theory 7(4), 985–1022 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. El Soufi, A., Ilias, S.: Immersions minimales, première valeur propre du laplacien et volume conforme. Math. Ann. 275(2), 257–267 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. El Soufi, A., Ilias, S.: Extremal metrics for the first eigenvalue of the Laplacian in a conformal class. Proc. Am. Math. Soc. 131(5), 1611–1618 (2003). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  19. El Soufi, A., Ilias, S., Ros, A.: Sur la première valeur propre des tores. In Séminaire de Théorie Spectrale et Géométrie, No. 15, Année 1996–1997, volume 15 of Sémin. Théor. Spectr. Géom., pp. 17–23. University of Grenoble I, Saint-Martin-d’Hères (1997)

  20. Gentile, G., Pagliara, V.: Riemannian metrics with large first eigenvalue on forms of degree \(p\). Proc. Am. Math. Soc. 123(12), 3855–3858 (1995)

    MathSciNet  MATH  Google Scholar 

  21. Girouard, A., Nadirashvili, N., Polterovich, I.: Maximization of the second positive Neumann eigenvalue for planar domains. J. Differ. Geom. 83(3), 637–661 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Grigor’yan, A., Netrusov, Y., Yau, S.-T.: Eigenvalues of elliptic operators and geometric applications. Surv. Differ. Geom. IX, 147–217 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hassannezhad, A.: Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem. J. Funct. Anal. 261(12), 3419–3436 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  25. Jammes, P.: Prescription de la multiplicité des valeurs propres du laplacien de Hodge-de Rham. Comment. Math. Helv. 86(4), 967–984 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jammes, P.: Spectre et géométrie conforme des variétés compactes à bord. Compos. Math. 150(12), 2112–2126 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jammes, P.: Une inégalité de Cheeger pour le spectre de Steklov. Ann. Inst. Fourier (Grenoble) 65(3), 1381–1385 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kokarev, G.: Variational aspects of Laplace eigenvalues on Riemannian surfaces. Adv. Math. 258, 191–239 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Korevaar, N.: Upper bounds for eigenvalues of conformal metrics. J. Differ. Geom. 37(1), 73–93 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kröger, P.: Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space. J. Funct. Anal. 106(2), 353–357 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Laugesen, R.S.: Eigenvalues of the Laplacian on inhomogeneous membranes. Am. J. Math. 120(2), 305–344 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. Milman, E.: On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177(1), 1–43 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Petrides, R.: Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces. Geom. Funct. Anal. 24(4), 1336–1376 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Petrides, R.: On a rigidity result for the first conformal eigenvalue of the Laplacian. J. Spectr. Theory 5(1), 227–234 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Schoen, R., Yau, S.-T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA (1994)

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Université de Neuchâtel, 13 rue E. Argand, 2007, Neuchâtel, Switzerland

    Bruno Colbois

  2. Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 7350, Université de Tours, Parc de Grandmont, 37200, Tours, France

    Ahmad El Soufi

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  1. Bruno Colbois
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  2. Ahmad El Soufi
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Correspondence to Bruno Colbois.

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Colbois, B., El Soufi, A. Spectrum of the Laplacian with weights. Ann Glob Anal Geom 55, 149–180 (2019). https://doi.org/10.1007/s10455-018-9621-5

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  • DOI: https://doi.org/10.1007/s10455-018-9621-5

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