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http://dx.doi.org/10.4134/CKMS.c180006

EIGENVALUE MONOTONICITY OF (p, q)-LAPLACIAN ALONG THE RICCI-BOURGUIGNON FLOW  

Azami, Shahroud (Department of Mathematics Faculty of Sciences Imam Khomeini International University)
Publication Information
Communications of the Korean Mathematical Society / v.34, no.1, 2019 , pp. 287-301 More about this Journal
Abstract
In this paper we study monotonicity the first eigenvalue for a class of (p, q)-Laplace operator acting on the space of functions on a closed Riemannian manifold. We find the first variation formula for the first eigenvalue of a class of (p, q)-Laplacians on a closed Riemannian manifold evolving by the Ricci-Bourguignon flow and show that the first eigenvalue on a closed Riemannian manifold along the Ricci-Bourguignon flow is increasing provided some conditions. At the end of paper, we find some applications in 2-dimensional and 3-dimensional manifolds.
Keywords
Laplace; Ricci-Bourguignon flow; eigenvalue;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 P. F. Leung, On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere, J. Austral. Math. Soc. Ser. A 50 (1991), no. 3, 409-416.   DOI
2 J.-F. Li, Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann. 338 (2007), no. 4, 927-946.   DOI
3 R. Manasevich and J. Mawhin, The spectrum of p-Laplacian systems with various boundary conditions and applications, Adv. Differential Equations 5 (2000), no. 10-12, 1289-1318.
4 A. Mukherjea and K. Pothoven, Real and Functional Analysis, Plenum Press, New York, 1978.
5 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv preprint math, 0211159, 2002.
6 J. Y.Wu, First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 8, 1591-1598.   DOI
7 H. Amann, Lusternik-Schnirelman theory and non-linear eigenvalue problems, Math. Ann. 199 (1972), 55-72.   DOI
8 S. Azami, The first eigenvalue of some (p, g)-Laplacian and geometric estimates, Commun. Korean Math. Soc. 33 (2018), no. 1, 317-323.   DOI
9 L. Boccardo and D. Guedes de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 3, 309-323.   DOI
10 J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global differential geometry and global analysis (Berlin, 1979), 42-63, Lecture Notes in Math., 838, Springer, Berlin, 1981.
11 X. Cao, Eigenvalues of $(-{\Delta}+{\frac{R}{2}})$ on manifolds with nonnegative curvature operator, Math. Ann. 337 (2007), no. 2, 435-441.   DOI
12 X. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc. 136 (2008), no. 11, 4075-4078.   DOI
13 P. L. de Napoli and M. C. Mariani, Quasilinear elliptic systems of resonant type and nonlinear eigenvalue problems, Abstr. Appl. Anal. 7 (2002), no. 3, 155-167.   DOI
14 G. Catino, L. Cremaschi, Z. Djadli, C. Mantegazza, and L. Mazzieri, The Ricci-Bourguignon flow, Pacific J. Math. 287 (2017), no. 2, 337-370.   DOI
15 B. Chen, Q. He, and F. Zeng, Monotonicity of eigenvalues of geometric operators along the Ricci-Bourguignon flow, Pacific J. Math. 296 (2018), no. 1, 1-20.   DOI
16 S. Y. Cheng, Eigenfunctions and eigenvalues of Laplacian, in Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2, 185-193, Amer. Math. Soc., Providence, RI, 1975.
17 Q.-M. Cheng and H. Yang, Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005), no. 2, 445-460.   DOI
18 J. I. Diaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. I, Research Notes in Mathematics, 106, Pitman (Advanced Publishing Program), Boston, MA, 1985.
19 P. L. de Napoli and J. P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, J. Differential Equations 227 (2006), no. 1, 102-115.   DOI
20 L. F. Di Cerbo, Eigenvalues of the Laplacian under the Ricci flow, Rend. Mat. Appl. (7) 27 (2007), no. 2, 183-195.
21 E. M. Harrell, II and P. L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Comm. Partial Differential Equations 19 (1994), no. 11-12, 2037-2055.   DOI
22 A. El Khalil, S. El Manouni, and M. Ouanan, Simplicity and stability of the first eigenvalue of a nonlinear elliptic system, Int. J. Math. Math. Sci. 2005 (2005), no. 10, 1555-1563.   DOI
23 D. A. Kandilakis, M. Magiropoulos, and N. B. Zographopoulos, The first eigenvalue of p-Laplacian systems with nonlinear boundary conditions, Bound. Value Probl. 2005 (2005), no. 3, 307-321.
24 A. El Khalil, Autour de la premiere courbe propre du p-Laplacien, These de Doctorat, 1999.