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Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

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Abstract

We obtain eigenvalue estimates for a larger class of elliptic differential operators in divergence form on a bounded domain in a complete Riemannian manifold isometrically immersed in Euclidean space. As an application, we give eigenvalue estimates in the Gaussian shrinking soliton, and we find a domain that makes the behavior of these estimates similar to the estimates for the case of the Laplacian. Moreover, we also give an answer to the generalized conjecture of Pólya.

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References

  1. Araújo Filho, M.C.: Inequalities for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds. Z. Angew. Math. Phys. 73(2), 1–28 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araújo Filho, M. C., Gomes, J. N. V.: Estimates of eigenvalues of an elliptic differential system in divergence form, To appear in Z. Angew. Math. Phys. (2022)

  3. Ashbaugh, M.S.: The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and HC Yang. Proc. Indian Acad. Sci. Math. Sci. 112(1), 3–30 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chavel, I.: Eigenvalues in Riemannian Geometry. Academic press, Cambridge (1984)

    MATH  Google Scholar 

  5. Chen, D., Cheng, Q.-M.: Extrinsic estimates for eigenvalues of the Laplace operator. J. Math. Soc. Jpn. 60(2), 325–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, Q.-M., Yang, H.: Estimates on eigenvalues of Laplacian. Math. Ann. 331(2), 445–460 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, Q.-M., Yang, H.: Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces. J. Math. Soc. Jpn. 58(2), 545–561 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheng, Q.-M., Yang, H.: Bounds on eigenvalues of Dirichlet Laplacian. Math. Ann. 337(1), 159–175 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng, Q.-M., Yang, H.: Estimates for eigenvalues on Riemannian manifolds. J. Differ. Equ. 247(8), 2270–2281 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheng, S.-Y., Yau, S.-T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225(3), 195–204 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics, vol 19, Instructor 67 (2009)

  12. Fonseca, J. C. M., Gomes, J. N. V.: Eigenvalue estimates of the drifted Cheng-Yau operator on bounded domains in pinched Cartan-Hadamard manifolds. arXiv:2107.09135v4 [math.DG]

  13. Gomes, J.N.V., Miranda, J.F.R.: Eigenvalue estimates for a class of elliptic differential operators in divergence form. Nonlinear Anal. 176, 1–19 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grebenkov, D.S., Nguyen, B.-T.: Geometrical structure of Laplacian eigenfunctions. SIAM Rev. 55(4), 601–667 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Harrell, E.M., II., Stubbe, J.: On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Am. Math. Soc. 349, 5 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hile, G.N., Protter, M.H.: Inequalities for eigenvalues of the Laplacian. Indiana Univ. Math. J. 29(4), 523–538 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. Levitin, M., Parnovski, L.: Commutators, spectral trance identities, and universal estimates for eigenvalues. J. Funct. Anal. 192, 425–445 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, P., Yau, S.-T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88(3), 309–318 (1983)

    Article  MATH  Google Scholar 

  19. Payne, L.E., Pólya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35(1–4), 289–298 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pólya, G.: On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc. 11(3), 419–433 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  21. Serre, D.: Divergence-free positive symmetric tensors and fluid dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(5), 1209–1234 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Weyl, H.: Über die asymptotische Verteilung der Eigenwerte. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse 1911, 110–117 (1911)

    MATH  Google Scholar 

  23. Xia, C., Xu, H.: Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds. Ann. Global Anal. Geom. 45(3), 155–166 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, H.: An estimate of the difference between consecutive eigenvalues, preprint IC/91/60 of ICTP, Trieste (1991)

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Acknowledgements

The first author has been partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) in conjunction with Fundação Rondônia de Amparo ao Desenvolvimento das Ações Científicas e Tecnológicas e à Pesquisa do Estado de Rondônia (FAPERO). The second author has been partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), of the Ministry of Science, Technology and Innovation of Brazil. Moreover, we would like to express our sincere thanks to the anonymous referee for his/her careful reading and useful comments which helped us improve our paper.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Rondônia, Campus Ji-Paraná R. Rio Amazonas, 351, Jardim dos Migrantes, Ji-Paraná, Rondônia, 76900-726, Brazil

    Marcio C. Araújo Filho

  2. Departamento de Matemática, Universidade Federal de São Carlos, Rod. Washington Luíz, Km 235, São Carlos, São Paulo, 13565-905, Brazil

    José N. V. Gomes

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  1. Marcio C. Araújo Filho
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  2. José N. V. Gomes
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All authors wrote and reviewed the main manuscript.

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Correspondence to Marcio C. Araújo Filho.

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Araújo Filho, M.C., Gomes, J.N.V. Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space. Z. Angew. Math. Phys. 74, 157 (2023). https://doi.org/10.1007/s00033-023-02054-1

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  • DOI: https://doi.org/10.1007/s00033-023-02054-1

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