In this paper, we prove that several different definitions of the Finsler-Laplacian are equivalent. Then we prove that any Berwald metric is affinely equivalent to its mean metric and give some upper or lower bound es...In this paper, we prove that several different definitions of the Finsler-Laplacian are equivalent. Then we prove that any Berwald metric is affinely equivalent to its mean metric and give some upper or lower bound estimates for the first eigenvalue of the mean Laplacian on Berwald manifolds, which generalize some results in Riemannian geometry.展开更多
This paper discusses the first eigenvalue on a compact Riemann manifold with the negative lower bound Ricci curvature. Let M be a compact Riemann manifold with the Ricci curvature≥-R, R=const. ≥0 and d is the diamet...This paper discusses the first eigenvalue on a compact Riemann manifold with the negative lower bound Ricci curvature. Let M be a compact Riemann manifold with the Ricci curvature≥-R, R=const. ≥0 and d is the diameter of M. Our main result is that the first eigenvalue λ1 of M satisfies λ1≥π^2/d^2-0.518R.展开更多
In this paper,we give a slight improvement of El Soufi–Ilias–Ros’s upper bound of the first Laplace eigenvalue on a torus in a fixed conformal class.We also optimize Montiel–Ros’s argument to obtain a better uppe...In this paper,we give a slight improvement of El Soufi–Ilias–Ros’s upper bound of the first Laplace eigenvalue on a torus in a fixed conformal class.We also optimize Montiel–Ros’s argument to obtain a better upper bound of the conformal area for certain rectangular tori.展开更多
Some complete variational formulas and approximation theorems for the first eigenvalue of elliptic operators in dimension one or a class of Markov chains are presented.
By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in ...By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.展开更多
In this paper, we present an explicit and computable lower bound for the first eigenvalue of birth-death processes with killing. This estimate is qualitatively sharp for birth-death processes without killing. We also ...In this paper, we present an explicit and computable lower bound for the first eigenvalue of birth-death processes with killing. This estimate is qualitatively sharp for birth-death processes without killing. We also establish an approximation procedure for the first eigenvalue of the birth-death process with killing by an increasing principal eigenvalue sequence of some birth-death processes without killing. Some applications of our results are illustrated by many examples.展开更多
We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature,if its first closed(resp.Neumann)eigenvalue of Finsler-Laplacian attains the sharp lower bound,then M is isometric to a circle(r...We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature,if its first closed(resp.Neumann)eigenvalue of Finsler-Laplacian attains the sharp lower bound,then M is isometric to a circle(resp.a segment).Moreover,a lower bound of the first eigenvalue of Finsler-Laplacian with Dirichlet boundary condition is also estimated.These generalize the corresponding results in recent literature.展开更多
In the paper we first derive theevolution equation for eigenvalues of geomet-ric operator-ΔФ+cR under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M,where,is the Witten-Laplacian oper...In the paper we first derive theevolution equation for eigenvalues of geomet-ric operator-ΔФ+cR under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M,where,is the Witten-Laplacian operator,Ф∈C^(∞)(M),and R is the scalar curvature.We then prove that the first eigenvalue of the geometricoperator is nondecreasing along the Ricci flow on closed surfaces with certain curva-ture conditions when 0<c≤1/2.As an application,we obtain some monotonicityformulae and estimates for the first eigenvalue on closed surfaces.展开更多
In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator （p ≥2） along the Ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator i...In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator （p ≥2） along the Ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator is nondecreasing along the Ricci flow under some different curvature assumptions, and therefore extend some parts of Ma＇s results [Ann. Glob. Anal.Geom,29,287-292（2006）]展开更多
In this paper, we study upper bounds of the first eigenvalue of a complete noncompact submanifold in an （n ＋p）-dimensional hyperbolic space H^n＋p. In particular, we prove that the first eigenvalue of a complete su...In this paper, we study upper bounds of the first eigenvalue of a complete noncompact submanifold in an （n ＋p）-dimensional hyperbolic space H^n＋p. In particular, we prove that the first eigenvalue of a complete submanifold in H^n＋p with parallel mean curvature vector H and finite L^q（q ≥ n） norm of traceless second fundamental form is not more than （n-1）2（1-（H）^2）/4 . We also prove that the first eigenvalue of a complete hypersurfaces which has finite index in H^n＋1（n ≤ 5） with constant mean curvature vector H and finite.展开更多
In this paper, the author has given an existence theorem for the resonant equation,-△pu=λ1｜u｜p-2u＋f（u）＋h（x）,without any Landesman-Lazer conditions on h（x）.
This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C^(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed e...This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C^(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed eigenvalue of the p-Laplacian on a compact Finsler manifold with nonnegative weighted Ricci curvature,on which a lower bound of the first Dirichlet eigenvalue of the p-Laplacian is also obtained.展开更多
We consider a parametric Dirichlet problem driven by the p-Laplacian with a Caratheodory reaction of equidiffusive type. Our hypotheses incorporate as a special case the equidiffusive p-logistic equation. We show that...We consider a parametric Dirichlet problem driven by the p-Laplacian with a Caratheodory reaction of equidiffusive type. Our hypotheses incorporate as a special case the equidiffusive p-logistic equation. We show that if λ1 〉 0 is the principal eigenvalue of the Dirichlet negative p-Laplacian and ）λ 〉 λ1 （/k being the parameter）, the problem has a unique positive solution, while for ）λ ∈ （0, λ1], the problem has no positive solution.展开更多
Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative co...Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau's estimation of the first eigenvalue is also given.展开更多
Constructing some proper functional spaces, we obtain the corresponding norm for the operator （-.L）^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Fr...Constructing some proper functional spaces, we obtain the corresponding norm for the operator （-.L）^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Front. Math. China, 2010, 5（3）： 379-515], for transient birth-death processes.展开更多
In this paper, we study a class of fourth-order Neumann boundary value problem (NBVP for short). By virtue of fixed point index and the spectral theory of linear operators, the existence of positive solutions is obtai...In this paper, we study a class of fourth-order Neumann boundary value problem (NBVP for short). By virtue of fixed point index and the spectral theory of linear operators, the existence of positive solutions is obtained under the assumption that the nonlinearity satisfies sublinear or superlinear conditions, which are relevant to the first eigenvalue of the corresponding linear operator.展开更多
In this paper, we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem of some doubly degenerate nonlinear parabolic equations.
We study the existence of positive solutions to boundary value problems for one- dimensional p-Laplacian under some conditions about the first eigenvalues correspon- ding to the relevant operators by the fixed point t...We study the existence of positive solutions to boundary value problems for one- dimensional p-Laplacian under some conditions about the first eigenvalues correspon- ding to the relevant operators by the fixed point theory. The main difficulties are the computation of fixed point index and the subadditivity for positively 1-homogeneous operator.展开更多
By virtue of the fixed point indices, we discuss the existence of the multiple positive periodic solutions to singular differential equation with state-dependent delay under the conditions concerning the first eigenva...By virtue of the fixed point indices, we discuss the existence of the multiple positive periodic solutions to singular differential equation with state-dependent delay under the conditions concerning the first eigenvalue of the relevant linear operator. The results in this paper are optimal and totally generalize many present results.展开更多
文摘In this paper, we prove that several different definitions of the Finsler-Laplacian are equivalent. Then we prove that any Berwald metric is affinely equivalent to its mean metric and give some upper or lower bound estimates for the first eigenvalue of the mean Laplacian on Berwald manifolds, which generalize some results in Riemannian geometry.
基金Supported by the NNSF of China(10271011)Supported by LIMB of the Ministry of Education China
文摘This paper discusses the first eigenvalue on a compact Riemann manifold with the negative lower bound Ricci curvature. Let M be a compact Riemann manifold with the Ricci curvature≥-R, R=const. ≥0 and d is the diameter of M. Our main result is that the first eigenvalue λ1 of M satisfies λ1≥π^2/d^2-0.518R.
文摘In this paper,we give a slight improvement of El Soufi–Ilias–Ros’s upper bound of the first Laplace eigenvalue on a torus in a fixed conformal class.We also optimize Montiel–Ros’s argument to obtain a better upper bound of the conformal area for certain rectangular tori.
基金This work was supported in part bythe National Natural Science Foundation of China (Grant No. 19631060) Mathematical Tian Yuan Foundation, Qiu Shi Science & Technology Foundation, RFDP and MCEC.
文摘Some complete variational formulas and approximation theorems for the first eigenvalue of elliptic operators in dimension one or a class of Markov chains are presented.
基金Supported by Foundation of Fujian’s Ministry of Education (Grant Nos. JA10058 and JA11051)National Natural Science Foundation of China (Grant No. 11126350)
文摘By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.
文摘In this paper, we present an explicit and computable lower bound for the first eigenvalue of birth-death processes with killing. This estimate is qualitatively sharp for birth-death processes without killing. We also establish an approximation procedure for the first eigenvalue of the birth-death process with killing by an increasing principal eigenvalue sequence of some birth-death processes without killing. Some applications of our results are illustrated by many examples.
基金supported by National Natural Science Foundation of China(Grant No.11171253)the Natural Science Foundation of Ministry of Education of Anhui Province(Grant No.KJ2012B197)
文摘We prove that for a compact Finsler manifold M with nonnegative weighted Ricci curvature,if its first closed(resp.Neumann)eigenvalue of Finsler-Laplacian attains the sharp lower bound,then M is isometric to a circle(resp.a segment).Moreover,a lower bound of the first eigenvalue of Finsler-Laplacian with Dirichlet boundary condition is also estimated.These generalize the corresponding results in recent literature.
基金PRC Grant NSFC(11371310,11401514,11471145)the University Science Research Project of Jiangsu Province(13KJB110029)+2 种基金the NaturalScience Foundation of Jiangsu Province(BK20140804)the Fundamental Research Funds for the CentralUniversities(NS2014076)Qing Lan Project.
文摘In the paper we first derive theevolution equation for eigenvalues of geomet-ric operator-ΔФ+cR under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M,where,is the Witten-Laplacian operator,Ф∈C^(∞)(M),and R is the scalar curvature.We then prove that the first eigenvalue of the geometricoperator is nondecreasing along the Ricci flow on closed surfaces with certain curva-ture conditions when 0<c≤1/2.As an application,we obtain some monotonicityformulae and estimates for the first eigenvalue on closed surfaces.
基金Supported by National Natural Science Foundation of China （Grant No. 10871069）
文摘In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator （p ≥2） along the Ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator is nondecreasing along the Ricci flow under some different curvature assumptions, and therefore extend some parts of Ma＇s results [Ann. Glob. Anal.Geom,29,287-292（2006）]
基金Supported by the National Natural Science Foundation of China(Grant No.11261038)the Natural Science Foundation of Jiangxi Province(Grant Nos.2010GZS0149+1 种基金20132BAB201005)Youth Science Foundation of Eduction Department of Jiangxi Province(Grant No.GJJ11044)
文摘In this paper, we study upper bounds of the first eigenvalue of a complete noncompact submanifold in an （n ＋p）-dimensional hyperbolic space H^n＋p. In particular, we prove that the first eigenvalue of a complete submanifold in H^n＋p with parallel mean curvature vector H and finite L^q（q ≥ n） norm of traceless second fundamental form is not more than （n-1）2（1-（H）^2）/4 . We also prove that the first eigenvalue of a complete hypersurfaces which has finite index in H^n＋1（n ≤ 5） with constant mean curvature vector H and finite.
基金Foundation item： Supported by the Sichuan Educational Comittee Science Foundation for Youths（08ZB002） Supported by the National Secience Foundation of Yibin University（2008Z02）
文摘In this paper, the author has given an existence theorem for the resonant equation,-△pu=λ1｜u｜p-2u＋f（u）＋h（x）,without any Landesman-Lazer conditions on h（x）.
基金supported by National Natural Science Foundation of China (Grant No. 11471246)Natural Science Foundation of Higher Education in Anhui Province (Grant No. KJ2014A257)
文摘This paper proves that the first eigenfunctions of the Finsler p-Lapalcian are C^(1,α). Using a gradient comparison theorem and one-dimensional model, we obtain the sharp lower bound of the first Neumann and closed eigenvalue of the p-Laplacian on a compact Finsler manifold with nonnegative weighted Ricci curvature,on which a lower bound of the first Dirichlet eigenvalue of the p-Laplacian is also obtained.
基金supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No.295118the National Science Center of Poland under grant No.N N201 604640+1 种基金the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under grant No.W111/7.PR/2012the National Science Center of Poland under Maestro Advanced Project No.DEC2012/06/A/ST1/00262
文摘We consider a parametric Dirichlet problem driven by the p-Laplacian with a Caratheodory reaction of equidiffusive type. Our hypotheses incorporate as a special case the equidiffusive p-logistic equation. We show that if λ1 〉 0 is the principal eigenvalue of the Dirichlet negative p-Laplacian and ）λ 〉 λ1 （/k being the parameter）, the problem has a unique positive solution, while for ）λ ∈ （0, λ1], the problem has no positive solution.
基金supported by the National Natural Science Foundation of China(Nos.11471246,11171253)the Natural Science Foundation of the Anhui Higher Education Institutions(No.KJ2014A257)
文摘Cheng-type inequality, Cheeger-type inequality and Faber-Krahn-type inequality are generalized to Finsler manifolds. For a compact Finsler manifold with the weighted Ricci curvature bounded from below by a negative constant, Li-Yau's estimation of the first eigenvalue is also given.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11101040, 11131003）, the 985 Project, the 973 Project （No. 2011CB808000）, the Specialized Research Fund for the Doctoral Program of Higher Education （No. 20100003110005）, and the Fundamental Research Funds for the Central Universities.
文摘Constructing some proper functional spaces, we obtain the corresponding norm for the operator （-.L）^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Front. Math. China, 2010, 5（3）： 379-515], for transient birth-death processes.
基金Supported by NNSF of China (No.60665001) Educational Department of Jiangxi Province(No.GJJ08358+1 种基金 No.GJJ08359 No.JXJG07436)
文摘In this paper, we study a class of fourth-order Neumann boundary value problem (NBVP for short). By virtue of fixed point index and the spectral theory of linear operators, the existence of positive solutions is obtained under the assumption that the nonlinearity satisfies sublinear or superlinear conditions, which are relevant to the first eigenvalue of the corresponding linear operator.
基金the Natural Science Foundation of Fujian Province of China (No.Z0511048)
文摘In this paper, we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem of some doubly degenerate nonlinear parabolic equations.
基金supported by the National Natural Science Foundation of China(11371221,11071141)the Specialized Research Foundation for the Doctoral Program of Higher Educationof China(20123705110001)the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province and Foundation of SDUST
文摘We study the existence of positive solutions to boundary value problems for one- dimensional p-Laplacian under some conditions about the first eigenvalues correspon- ding to the relevant operators by the fixed point theory. The main difficulties are the computation of fixed point index and the subadditivity for positively 1-homogeneous operator.
基金Supported by National Natural Science Foundation of China (No.10626029, No.10701040)Edu-cational Department of Jiangxi Province (No.GJJ08358, No.GJJ08359, No.JXJG07436)Jiangxi University of Finance and Economics (No.04232015, No.JXCDJG0813).
文摘By virtue of the fixed point indices, we discuss the existence of the multiple positive periodic solutions to singular differential equation with state-dependent delay under the conditions concerning the first eigenvalue of the relevant linear operator. The results in this paper are optimal and totally generalize many present results.
