March  2017, 6(2): 475-491. doi: 10.3934/cpaa.2017024

Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian

1. 

School of Mathematics and Physics Science, Jingchu University of Technology, Jingmen, 448000, China

2. 

Departamento de Matemática, Universidade de Brasilia, 70910-900-Brasilia-DF, Brazil

Received  April 2016 Revised  October 2016 Published  January 2016

In this paper, we firstly study the eigenvalue problem of a systemof elliptic equations with drift and get some universal inequalities of PayneP′olya-Weinberger-Yang type on a bounded domain in Euclidean spaces and inGaussian shrinking solitons. Furthermore, we study two kinds of the clampedplate problems and the buckling problems for the bi-drifting Laplacian and getsome sharp lower bounds for the first eigenvalue for these eigenvalue problemon compact manifolds with boundary and positive m-weighted Ricci curvatureor on compact manifolds with boundary under some condition on the weightedRicci curvature.

Citation: Feng Du, Adriano Cavalcante Bezerra. Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian. Communications on Pure and Applied Analysis, 2017, 6 (2) : 475-491. doi: 10.3934/cpaa.2017024
References:
[1]

D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math. , pages 177-206. Springer, Berlin, 1985. doi: 10.1007/BFb0075847.

[2]

M. Batista, M. P. Cavalcante and J. Pyo, Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds, J. Math. Anal. Appl., 419 (2014), 617-626. doi: 10.1016/j.jmaa.2014.04.074.

[3]

D. Chen, Q. M. Cheng, Q. Wang and C. Xia, On eigenvalues of a system of elliptic equations and of the biharmonic operator, J. Math. Anal. Appl., 387 (2012), 1146-1159. doi: 10.1016/j.jmaa.2011.10.020.

[4]

X. Cheng, T. Mejia and D. Zhou, Eigenvalue estimate and compactness for closed f-minimal surfaces, Pacific J. Math., 271 (2014), 347-367. doi: 10.2140/pjm.2014.271.347.

[5]

Q. M. Cheng and H. C. Yang, Universal inequalities for eigenvalues of a system of elliptic equations, Proc. Royal Soc. Edinburgh, 139A (2009), 273-285. doi: 10.1017/S0308210507000649.

[6]

F. Du, C. Wu, G. Li and C. Xia, Universal inequalities for eigenvalues of a system of subelliptic equations on Heisenberg group, Kodai Math. J., 38 (2015), 437-450. doi: 10.2996/kmj/1436403899.

[7]

F. Du, C. Wu, G. Li and C. Xia, Estimates for eigenvalues of the bi-drifting Laplacian operator, Z. Angew. Math. Phys., 66 (2015), 703-726. doi: 10.1007/s00033-014-0426-5.

[8]

A. Futaki, H. Li and X. D. Li, On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons, Ann. Glob. Anal. Geom., 44 (2013), 105-114. doi: 10.1007/s10455-012-9358-5.

[9]

Q. Huang and Q. H. Ruan, Applications of some elliptic equations in Riemannian manifolds, J. Math. Anal. Appl., 409 (2014), 189-196. doi: 10.1016/j.jmaa.2013.07.004.

[10]

S. M. Hook, Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc., 318 (1990), 615-642. doi: 10.2307/2001323.

[11]

H. Li and Y. Wei, f-minimal surface and manifold with positive m-Bakry-Émery Ricci curvature, J. Geom. Anal., 25 (2015), 421-435. doi: 10.1007/s12220-013-9434-5.

[12]

H. A. Levine and M. H. Protter, Unretricted lower bounds for eigenvalues of elliptic equations and systems of equations with applications to problem in elasticity, Math. Methods Appl. Sci., 7 (1985), 210-222. doi: 10.1002/mma.1670070113.

[13]

M. Levitin and L. Parnovski, Commutators, spectral trance identities, and universal estimates for eigenvalues, J. Funct. Anal., 192 (2002), 425-445. doi: 10.1006/jfan.2001.3913.

[14]

L. Ma and S. H. Du, Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians, C. R. Math. Acad. Sci. Paris., 348 (2010), 1203-1206. doi: 10.1016/j.crma.2010.10.003.

[15]

L. Ma and B. Y. Liu, Convex eigenfunction of a drifting Laplacian operator and the fundamental gap, Pacific J. Math., 240 (2009), 343-361. doi: 10.2140/pjm.2009.240.343.

[16]

L. Ma and B. Y. Liu, Convexity of the first eigenfunction of the drifting Laplacian operator and its applications, New York J. Math., 14 (2008), 393-401.

[17]

R. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., 26 (1977), 459-472.

[18]

G. Wei and W. Wylie, Comparison geometry for the Bakry-Émery Ricci tensor, J. Diff. Geom., 83 (2009), 377-405.

[19]

C. Xia and H. Xu, Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds, Ann. Glob. Anal. Geom., 45 (2014), 155-166. doi: 10.1007/s10455-013-9392-y.

show all references

References:
[1]

D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math. , pages 177-206. Springer, Berlin, 1985. doi: 10.1007/BFb0075847.

[2]

M. Batista, M. P. Cavalcante and J. Pyo, Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds, J. Math. Anal. Appl., 419 (2014), 617-626. doi: 10.1016/j.jmaa.2014.04.074.

[3]

D. Chen, Q. M. Cheng, Q. Wang and C. Xia, On eigenvalues of a system of elliptic equations and of the biharmonic operator, J. Math. Anal. Appl., 387 (2012), 1146-1159. doi: 10.1016/j.jmaa.2011.10.020.

[4]

X. Cheng, T. Mejia and D. Zhou, Eigenvalue estimate and compactness for closed f-minimal surfaces, Pacific J. Math., 271 (2014), 347-367. doi: 10.2140/pjm.2014.271.347.

[5]

Q. M. Cheng and H. C. Yang, Universal inequalities for eigenvalues of a system of elliptic equations, Proc. Royal Soc. Edinburgh, 139A (2009), 273-285. doi: 10.1017/S0308210507000649.

[6]

F. Du, C. Wu, G. Li and C. Xia, Universal inequalities for eigenvalues of a system of subelliptic equations on Heisenberg group, Kodai Math. J., 38 (2015), 437-450. doi: 10.2996/kmj/1436403899.

[7]

F. Du, C. Wu, G. Li and C. Xia, Estimates for eigenvalues of the bi-drifting Laplacian operator, Z. Angew. Math. Phys., 66 (2015), 703-726. doi: 10.1007/s00033-014-0426-5.

[8]

A. Futaki, H. Li and X. D. Li, On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons, Ann. Glob. Anal. Geom., 44 (2013), 105-114. doi: 10.1007/s10455-012-9358-5.

[9]

Q. Huang and Q. H. Ruan, Applications of some elliptic equations in Riemannian manifolds, J. Math. Anal. Appl., 409 (2014), 189-196. doi: 10.1016/j.jmaa.2013.07.004.

[10]

S. M. Hook, Domain independent upper bounds for eigenvalues of elliptic operator, Trans. Amer. Math. Soc., 318 (1990), 615-642. doi: 10.2307/2001323.

[11]

H. Li and Y. Wei, f-minimal surface and manifold with positive m-Bakry-Émery Ricci curvature, J. Geom. Anal., 25 (2015), 421-435. doi: 10.1007/s12220-013-9434-5.

[12]

H. A. Levine and M. H. Protter, Unretricted lower bounds for eigenvalues of elliptic equations and systems of equations with applications to problem in elasticity, Math. Methods Appl. Sci., 7 (1985), 210-222. doi: 10.1002/mma.1670070113.

[13]

M. Levitin and L. Parnovski, Commutators, spectral trance identities, and universal estimates for eigenvalues, J. Funct. Anal., 192 (2002), 425-445. doi: 10.1006/jfan.2001.3913.

[14]

L. Ma and S. H. Du, Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians, C. R. Math. Acad. Sci. Paris., 348 (2010), 1203-1206. doi: 10.1016/j.crma.2010.10.003.

[15]

L. Ma and B. Y. Liu, Convex eigenfunction of a drifting Laplacian operator and the fundamental gap, Pacific J. Math., 240 (2009), 343-361. doi: 10.2140/pjm.2009.240.343.

[16]

L. Ma and B. Y. Liu, Convexity of the first eigenfunction of the drifting Laplacian operator and its applications, New York J. Math., 14 (2008), 393-401.

[17]

R. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., 26 (1977), 459-472.

[18]

G. Wei and W. Wylie, Comparison geometry for the Bakry-Émery Ricci tensor, J. Diff. Geom., 83 (2009), 377-405.

[19]

C. Xia and H. Xu, Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds, Ann. Glob. Anal. Geom., 45 (2014), 155-166. doi: 10.1007/s10455-013-9392-y.

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