黎曼流形上关于p-Laplacian的ν-Euclidean类型的Faber-Krahn不等式
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摘要
首先利用Federer-Fleming定理研究了黎曼流形上p-Laplace算子的解析Faber-Krahn不等式;其次利用余面积公式和Cavalieri原理研究了黎曼流形上p-Laplace算子的解析Faber-Krahn不等式的一般化.
The Federer-Fleming Theorem is firstly used to investigate the analytic Faber-Krahn inequalities of Euclidean type for the p-Laplace operator on manifolds. Secondly,the coarea formula and Cavalieri's principle is applied to study the general Faber-Krahn inequalities of Euclidean type for the p-Laplace operator on manifolds.
The Federer-Fleming Theorem is firstly used to investigate the analytic Faber-Krahn inequalities of Euclidean type for the p-Laplace operator on manifolds. Secondly,the coarea formula and Cavalieri's principle is applied to study the general Faber-Krahn inequalities of Euclidean type for the p-Laplace operator on manifolds.
引文
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[6]杨慧章.局部对称伪黎曼流形中具有平行平均曲率向量的类空子流形[J].安徽大学学报(自然科学版),2018,42(4):45-49.YANG Huizhang. Space-like submanifolds with parallel mean curvature in locally symmetric pseudo-Riemannian manifold[J].Journal of Anhui University(Natural Science Edition),2018,42(4):45-49.
[7]HUANG Guangyue,MA Bingqing. Eigenvalue estimates for submanifolds with bounded f-mean curvature[J]. Proceedings of the Indian Academy of Sciences Mathematical Sciences,2017,127(2):375-381.
[8]CHAVEL I. Isoperimetric inequalities[M]. Cambridge:Cambridge University Press,2001.
[9]FEDERER H,FLEMING W H. Normal integral currents[J]. Annals of Mathematics,1960,72(3):458-520.
[10]MAZ’YA V G. Classes of domains and embeding theorems for functional spaces[J]. Dokl Acad Nauk SSSR,1960,133:527-530.
[11]ADAMS R A. Soboleve spaces[M]. New York:Academic Press,1975:29.
[12]ROS A. Compact hypersurfaces with constant higher order mean curvatures[J]. Rev Mat Iberoamericana,1987,3(3):447-453.
[13]CASELLES V,CHAMBOLLE A,NOVAGA M. Uniqueness of the Cheeger set of a convex body[J]. Pacific J Math,2007,232(1):77-90.
[14]WEI S S,ZHU M J. Sharp isoperimetric inequalities and sphere theorems[J]. Pacific Journal of Mathematics,2005,220(1):183-195.
[15]YAU S T. Isoperimetric constants and the first eigenvalue of a compact manifold[J]. Ann Sci C Norm Sup,1975,8:487-507.
[16]GRIGOR’YAN A A. Heat kernel on a manifold with a local Harnack inequaity[J]. Anal Geom,1994,2(1):111-138.
[2]WANG Yuzhao,LI Huaiqian. Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces[J]. Differential Geometry and Its Applications,2016,45:23-42.
[3]DU Feng,MAO Jing. Estimates for the first eigenvalue of the drifting Laplace,and the p-Laplace operators on submanifolds with bounded mean curvature in the hyperbolic space[J]. Journal of Mathematical Analysis and Applications,2017,456(2):787-795.
[4]马丽,董唯光,梁金平,等.基于随机投影的正交判别流形学习算法[J].郑州大学学报(理学版),2016,48(1):102-109,115.MA Li DONG Weiguang LIANG Jinping,et al. Manifold learning algorithms of orthogonal discriminant based on random projection[J]. Journal of Zhengzhou University(Natural Science Edition),2016,48(1):102-109,115.
[5]潘虹,李静,张倩玉.非正截面曲率空间中F-双调和子流形的若干结果[J].信阳师范学院学报(自然科学版),2016,29(4):501-506.PAN Hong,LI Jing,ZHANG Qianyu. Some results of F-biharmonic submanifolds in the non-positive sectional curvature space[J]. Journal of Xinyang Normal University(Natural Science Edition),2016,29(4):501-506.
[6]杨慧章.局部对称伪黎曼流形中具有平行平均曲率向量的类空子流形[J].安徽大学学报(自然科学版),2018,42(4):45-49.YANG Huizhang. Space-like submanifolds with parallel mean curvature in locally symmetric pseudo-Riemannian manifold[J].Journal of Anhui University(Natural Science Edition),2018,42(4):45-49.
[7]HUANG Guangyue,MA Bingqing. Eigenvalue estimates for submanifolds with bounded f-mean curvature[J]. Proceedings of the Indian Academy of Sciences Mathematical Sciences,2017,127(2):375-381.
[8]CHAVEL I. Isoperimetric inequalities[M]. Cambridge:Cambridge University Press,2001.
[9]FEDERER H,FLEMING W H. Normal integral currents[J]. Annals of Mathematics,1960,72(3):458-520.
[10]MAZ’YA V G. Classes of domains and embeding theorems for functional spaces[J]. Dokl Acad Nauk SSSR,1960,133:527-530.
[11]ADAMS R A. Soboleve spaces[M]. New York:Academic Press,1975:29.
[12]ROS A. Compact hypersurfaces with constant higher order mean curvatures[J]. Rev Mat Iberoamericana,1987,3(3):447-453.
[13]CASELLES V,CHAMBOLLE A,NOVAGA M. Uniqueness of the Cheeger set of a convex body[J]. Pacific J Math,2007,232(1):77-90.
[14]WEI S S,ZHU M J. Sharp isoperimetric inequalities and sphere theorems[J]. Pacific Journal of Mathematics,2005,220(1):183-195.
[15]YAU S T. Isoperimetric constants and the first eigenvalue of a compact manifold[J]. Ann Sci C Norm Sup,1975,8:487-507.
[16]GRIGOR’YAN A A. Heat kernel on a manifold with a local Harnack inequaity[J]. Anal Geom,1994,2(1):111-138.