具有加权测度的H型群上漂移Laplace算子的Levitin-Parnovski型特征值不等式
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摘要
本文研究了具有加权测度dμ=e~(-φ)dv的H型群G上漂移Laplace算子-?_G+<▽_Gφ,▽_G(·)>的Dirichlet特征值问题,建立了该问题的Levitin-Parnovski型特征值不等式,推广包含了Ilias和Makhoul对Heisenberg群上次Laplace算子所获得的结果 (J. Geom. Anal.,2012, 22(1):206–222).
In this paper, we study the Dirichlet eigenvalue problem of the drifting Laplacian -?_G+<▽_Gφ,▽_G(·)>on the H-type group G with the weighted measured dμ = e~(-φ)dv. We establish a Levitin-Parnovski universal inequality for eigenvalues of this problem, which generalize the result derived by Ilias and Makhoul for the Kohn Laplacian on the Heisenberg group(J. Geom.Anal., 2012, 22(1): 206–222).
In this paper, we study the Dirichlet eigenvalue problem of the drifting Laplacian -?_G+<▽_Gφ,▽_G(·)>on the H-type group G with the weighted measured dμ = e~(-φ)dv. We establish a Levitin-Parnovski universal inequality for eigenvalues of this problem, which generalize the result derived by Ilias and Makhoul for the Kohn Laplacian on the Heisenberg group(J. Geom.Anal., 2012, 22(1): 206–222).
引文
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[2]Garofalo N,Vassilev D.Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians[J].Math.Ann.,2000,318(3):453-516.
[3]Garofalo N,Vassilev D.Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type[J].Duke.Math.J.,2001,106(3):411-448.
[4]Colding T H,Ii W P M.Generic mean curvature flow I;generic singularities[J].Ann.Math.,2009,2(2):755-833.
[5]Futaki A,Li H,Li X D.On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons[J].Ann.Global Anal.Geom.,2013,44(2):105-114.
[6]Huang G Y,Zhang C,Zhang J.Liouville-type theorem for the drifting Laplacian operator[J].Arch.Math.(Basel),2011,96(4):379-385.
[7]Friedrichs K O.Spectral theory of operators in Hilbert space[M].New York:Springer,1973.
[8]Inglis J.Spectral inequalities for operators on H-type groups[J].J.Spectr.Theory.,2012,2(1):79-105.
[9]Niu P C,Zhang H Q.Payne-P′olya-Weinberger type inequalities for eigenvalues of nonelliptic operator[J].Pac.J.Math.,2003,208(2):325-345.
[10]Sun H J.Universal inequalities and bounds for weighted eigenvalues of the Schr¨odinger operator on the Heisenberg group[J].Turkish J.Math.,2011(35):249-258.
[11]韩军强,钮鹏程.H型群上次Laplace算子相邻特征值之差的估计[J].西北大学学报(自然科学版),2006,36(4):522-524.
[12]谭沈阳,黄体仁.H型群上Witten-Laplace算子的特征值估计[J].数学进展,2016,45(3):417-423.
[13]Ilias S,Makhoul O.A generalization of a Levitin and Parnovski universal inequality for eigenvalues[J].J.Geom.Anal.,2012,22(1):206-222.
[14]Ashbaugh S M,Benguria D R.More bounds on eigenvalues ratios for Dirichlet Laplacians in n dimensions[J].Siam J.Math.Anal.,1993,24(6):1622-1651.
[15]Sun H J,Cheng Q M,Yang H C.Lower order eigenvalue of Dirichlet Laplacian[J].Manuscripta Math.,2008,125(2):139-156.
[16]Levitin M,Parnovski L.Commutators,spectral trace identities,and universal estimates for eigenvalues[J].J.Funct.Anal.,2002,192(2):425-445.
[2]Garofalo N,Vassilev D.Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians[J].Math.Ann.,2000,318(3):453-516.
[3]Garofalo N,Vassilev D.Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type[J].Duke.Math.J.,2001,106(3):411-448.
[4]Colding T H,Ii W P M.Generic mean curvature flow I;generic singularities[J].Ann.Math.,2009,2(2):755-833.
[5]Futaki A,Li H,Li X D.On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons[J].Ann.Global Anal.Geom.,2013,44(2):105-114.
[6]Huang G Y,Zhang C,Zhang J.Liouville-type theorem for the drifting Laplacian operator[J].Arch.Math.(Basel),2011,96(4):379-385.
[7]Friedrichs K O.Spectral theory of operators in Hilbert space[M].New York:Springer,1973.
[8]Inglis J.Spectral inequalities for operators on H-type groups[J].J.Spectr.Theory.,2012,2(1):79-105.
[9]Niu P C,Zhang H Q.Payne-P′olya-Weinberger type inequalities for eigenvalues of nonelliptic operator[J].Pac.J.Math.,2003,208(2):325-345.
[10]Sun H J.Universal inequalities and bounds for weighted eigenvalues of the Schr¨odinger operator on the Heisenberg group[J].Turkish J.Math.,2011(35):249-258.
[11]韩军强,钮鹏程.H型群上次Laplace算子相邻特征值之差的估计[J].西北大学学报(自然科学版),2006,36(4):522-524.
[12]谭沈阳,黄体仁.H型群上Witten-Laplace算子的特征值估计[J].数学进展,2016,45(3):417-423.
[13]Ilias S,Makhoul O.A generalization of a Levitin and Parnovski universal inequality for eigenvalues[J].J.Geom.Anal.,2012,22(1):206-222.
[14]Ashbaugh S M,Benguria D R.More bounds on eigenvalues ratios for Dirichlet Laplacians in n dimensions[J].Siam J.Math.Anal.,1993,24(6):1622-1651.
[15]Sun H J,Cheng Q M,Yang H C.Lower order eigenvalue of Dirichlet Laplacian[J].Manuscripta Math.,2008,125(2):139-156.
[16]Levitin M,Parnovski L.Commutators,spectral trace identities,and universal estimates for eigenvalues[J].J.Funct.Anal.,2002,192(2):425-445.