Abstract
There are many interesting eigenvalue problems in a variety of settings; one of them is the well-known Steklov eigenvalue problem. In this work, we are interested in studying some Steklov eigenvalue problems for elliptic operators of second and fourth order using a well-known Reilly formula. Some upper and lower bounds for the first eigenvalue are obtained, and the rigidity case is carefully analyzed.
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References
Bakry, D., Émery M.: Diffusions hypercontractives. Séminaire de probabilities, XIX, 1983/84, Lecture Notes in Math, 1123, Springer, Berlin, pp 177–206 (1985)
Batista, M., Cavalcante, M.P., Pyo, J.: Some Isoperimetric Inequalities and Eigenvalue Estimates in Weighted Manifolds. J. Math. Anal. Appl. 419, 617–626 (2014)
Binoy, R., Santhanam, G.: Sharp upperbound and a comparison theorem for the first nonzero Steklov eigenvalue. J. Ramanujan Math. Soc. 29(2), 133–154 (2014)
Escobar, J.F.: The geometry of the first non-zero Stekloff eigenvalue. J. Funct. Anal. 150(2), 544–556 (1997)
Escobar, J.F.: An isoperimetric inequality and the first Steklov eigenvalue. J. Funct. Anal. 165101–116 (1999)
Escobar, J.F.: A comparison theorem for the first non-zero Steklov eigenvalue. J. Funct. Anal. 178(1), 143–155 (2000)
Girouard, A., Polterovich, I.: Spectral Geometry of the Steklov problem. J. Spectral Theory 7321–359 (2017)
Gilbarg, D., Trudinger N.S.: Elliptic partial differential equations of second order. Reprint of the 1998 ed. Cassics in Mathematics. Springer-Verlag, Berlin, (2001)
Gromov, M.: Isoperimetric of waists and concentration of maps. Geom. Funct. Anal. 13(1), 178–215 (2003)
Huang, Q., Ruan, Q.: Application of Some Elliptic Equations in Riemannian Manifolds. J. Math. Anal. Appl. 409(1), 189–196 (2014)
Kuttler, J.R., Sigillito, V.G.: Inequalities for membrane and Stekloff eigenvalues. J. Math. Anal. Appl. 23, 148–160 (1968)
Lichnerowich, A.: Variétés riemanniennes á tenseur \(C\) non négatif. C. R. Acad. Sci. Paris Sér. A-B 271, A650–A653 (1970)
Ma, L., Du, S.H.: Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. C. R. Math. Acad. Sci. Paris 348(21–22), 1203–1206 (2010)
Payne, L.E.: Some isoperimetric inequalities for harmonic functions. SIAM J. Math. Anal. 1, 354–359 (1970)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint available at arXiv:math.DG0211159
Raulot, S., Savo, A.: On the first eigenvalue of the Dirichlet-to-Neumann operator on forms. J. Funct. Anal. 262(3), 889–914 (2012)
Reilly, R.: Geometric applications of the solvability of Neumann problems on a Riemannian manifold. Arch. Rational Mech. Anal. 75(1), 23–29 (1980)
Stekloff, M.W.: Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. École Norm. Sup. 19, 455–490 (1902)
Weinstock, R.: Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal. 3, 745–753 (1954)
Xia, C., Wang, Q.: Sharp bounds for the first non-zero Stekloff eigenvalues. J. Funct. Anal. 257(8), 2635–2644 (2009)
Acknowledgements
The authors would like to thank the referee for the valuable comments and suggestions, which helped us to improve and clarify the manuscript
Funding
The first author was partially supported by Alagoas Research Foundation [Grant: E:60030.0000001758/2022], and by the Brazilian National Council for Scientific and Technological Development, Brazil [Grant: 308440/2021-8 and 405468/2021-0], and both authors were partially supported by Coordination for the Improvement of Higher Education Personnel [Finance code - 001]
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Batista, M., Santos, J.I. Manifolds with Density and the First Steklov Eigenvalue. Potential Anal 60, 1369–1382 (2024). https://doi.org/10.1007/s11118-023-10091-8
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DOI: https://doi.org/10.1007/s11118-023-10091-8