International Journal of Nonlinear Analysis and Applications
Variation of the first eigenvalue of $(p,q)$-Laplacian along the Ricci-harmonic flow
Document Type : Research Paper
Author
Department of Pure Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran.
Abstract
In this paper, we study monotonicity for the first eigenvalue of a class of $(p,q)$-Laplacian. We find the first variation formula for the first eigenvalue of $(p,q)$-Laplacian on a closed Riemannian manifold evolving by the Ricci-harmonic flow and construct various monotonic quantities by imposing some conditions on initial manifold.
Keywords
[1] A. Abolarinwa, Evolution and monotonicity of the first eigenvalue of p-Laploacian under the Ricci-harmonic flow, J. Appl. Anal. 21 (2015) 147–160.
[2] X.-D. Cao, Eigenvalues of (−∆ + R2) on manifolds with nonnegative curvature operator, Math. Ann. 337(2) (2007) 435–447.
[3] X.-D, Cao, First eigenvalues of geometric operators under the Ricci flows, Proc. Amer. Math. Soc. 136 (2008)4075–4078.
[4] J. I. D´ıaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. I. Elliptic Equations, Research Notes in Mathematics, vol. 106, Pitman, Massachusetts, 1985.
[5] A. E. Khalil, Autour de la premi´ere courbe propre du p-Laplacien, Th´ese de Doctorat, 1999.
[6] A. E. Khalil, S. E. Manouni, M. Ouanan, Simplicity and stablity of the first eigenvalue of a nonlinear elliptic system, Int. J. Math. Math. Sci. 10 (2005) 1555–1563.
[7] J. F. Li, Eigenvalues and energy functionals with monotonicity formula under Ricci flow, Math. Ann. 338 (2007) 927–946.
[8] A. Mukherjea and K. Pothoven, Real and Functional Analysis, 2nd, Plenum Press, New York and London, 1984.
[9] R. M¨uller, Ricci flow coupled with harmonic map flow, Ann. Sci. de l’Ecole Normale Sup. 45 (2012) 101–142. ´
[10] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, (2002), ArXiv: 0211159.
[11] J. Y. Wu, First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow, Acta Math. Sinica, English Series, 27(8) (2011) 1591–1598.
[12] F. Zeng, Q. He and B. Chen, Monotonicity of eigenvalues of geometric operators along the Ricci-Bourguignon flow, Arxiv, 152.08158v1, 2016.
[13] L. Zhao, The first eigenvalue of the p-laplacin operator under powers of the mth mean curvature flow, Results Math. 63 (2012) 937–948.
[2] X.-D. Cao, Eigenvalues of (−∆ + R2) on manifolds with nonnegative curvature operator, Math. Ann. 337(2) (2007) 435–447.
[3] X.-D, Cao, First eigenvalues of geometric operators under the Ricci flows, Proc. Amer. Math. Soc. 136 (2008)4075–4078.
[4] J. I. D´ıaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. I. Elliptic Equations, Research Notes in Mathematics, vol. 106, Pitman, Massachusetts, 1985.
[5] A. E. Khalil, Autour de la premi´ere courbe propre du p-Laplacien, Th´ese de Doctorat, 1999.
[6] A. E. Khalil, S. E. Manouni, M. Ouanan, Simplicity and stablity of the first eigenvalue of a nonlinear elliptic system, Int. J. Math. Math. Sci. 10 (2005) 1555–1563.
[7] J. F. Li, Eigenvalues and energy functionals with monotonicity formula under Ricci flow, Math. Ann. 338 (2007) 927–946.
[8] A. Mukherjea and K. Pothoven, Real and Functional Analysis, 2nd, Plenum Press, New York and London, 1984.
[9] R. M¨uller, Ricci flow coupled with harmonic map flow, Ann. Sci. de l’Ecole Normale Sup. 45 (2012) 101–142. ´
[10] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, (2002), ArXiv: 0211159.
[11] J. Y. Wu, First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow, Acta Math. Sinica, English Series, 27(8) (2011) 1591–1598.
[12] F. Zeng, Q. He and B. Chen, Monotonicity of eigenvalues of geometric operators along the Ricci-Bourguignon flow, Arxiv, 152.08158v1, 2016.
[13] L. Zhao, The first eigenvalue of the p-laplacin operator under powers of the mth mean curvature flow, Results Math. 63 (2012) 937–948.
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Volume 12, Issue 2
November 2021Pages 193-204
- Receive Date: 22 July 2019
- Revise Date: 19 January 2020
- Accept Date: 22 January 2020