Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =(X1,X2,... ,Xm) be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmande...Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =(X1,X2,... ,Xm) be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander's condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.展开更多
By using diffusion process with absorbing boundary,some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+▽h on a non-compact complete Riemannian manifold. The resulting estimates contain Mc...By using diffusion process with absorbing boundary,some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+▽h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean’s estimate for ▽h=0.Moreover,the first Dirichlet eigen- value for elliptic operators on R^d and the first mixed eigenvalue are also studied.Some examples show that our estimates can be sharp even for ▽h≠0.展开更多
LetΩ,with finite Lebesgue measure|Ω|,be a non-empty open subset of R,andΩ=∪∞j-1Ωj,where the open setsΩj are pairwise disjoint and the boundaryГ=■Ωhas Minkowski dimension D∈(0,1).In this paper we study the D...LetΩ,with finite Lebesgue measure|Ω|,be a non-empty open subset of R,andΩ=∪∞j-1Ωj,where the open setsΩj are pairwise disjoint and the boundaryГ=■Ωhas Minkowski dimension D∈(0,1).In this paper we study the Dirichlet eigenvalues problem on the domainΩand give the exact second asymptotic term for the eigenvalues,which is related to the Minkowski dimension D.Meanwhile,we give sharp lower bound estimates for Dirichlet eigenvalues for such one-dimensional fractal domains.展开更多
In this paper,we study large m asymptotics of the l 1 minimal m-partition problem for the Dirichlet eigenvalue.For any smooth domainΩ⊂R^(n)such that|Ω|=1,we prove that the limit lim_(m→∞)l^(1)_(m)(Ω)=c 0 exists,a...In this paper,we study large m asymptotics of the l 1 minimal m-partition problem for the Dirichlet eigenvalue.For any smooth domainΩ⊂R^(n)such that|Ω|=1,we prove that the limit lim_(m→∞)l^(1)_(m)(Ω)=c 0 exists,and the constant c 0 is independent of the shape ofΩ.Here,l^(1)_(m)(Ω)denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any m-partition ofΩ.展开更多
The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upp...The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upper bound for its Dirichlet eigenvalues.展开更多
This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω , u = 0, x ∈δΩ, where Ω belong to R^n is a smooth bounded convex domain. By using the method of appropriate barr...This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω , u = 0, x ∈δΩ, where Ω belong to R^n is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T. Yau et al.展开更多
Simultaneous measurements of position and momentum are considered in n dimensions. We find, that for a particle whose position is strictly localized in a compact domain (spatial uncertainty) with non-empty boundary, t...Simultaneous measurements of position and momentum are considered in n dimensions. We find, that for a particle whose position is strictly localized in a compact domain (spatial uncertainty) with non-empty boundary, the standard deviation of its momentum is sharply bounded by , while is the first Dirichlet eigenvalue of the Laplacian on D.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
基金partially supported by the NSFC(11631011,11626251)
文摘Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =(X1,X2,... ,Xm) be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander's condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.
基金Research supported in part by NFSCthe State Education Commission of China
文摘By using diffusion process with absorbing boundary,some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+▽h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean’s estimate for ▽h=0.Moreover,the first Dirichlet eigen- value for elliptic operators on R^d and the first mixed eigenvalue are also studied.Some examples show that our estimates can be sharp even for ▽h≠0.
基金The research has been supported by the grants from National Natural Science Foundation of China(Grants No.11631011).
文摘LetΩ,with finite Lebesgue measure|Ω|,be a non-empty open subset of R,andΩ=∪∞j-1Ωj,where the open setsΩj are pairwise disjoint and the boundaryГ=■Ωhas Minkowski dimension D∈(0,1).In this paper we study the Dirichlet eigenvalues problem on the domainΩand give the exact second asymptotic term for the eigenvalues,which is related to the Minkowski dimension D.Meanwhile,we give sharp lower bound estimates for Dirichlet eigenvalues for such one-dimensional fractal domains.
基金supported by National Science Foundation of USA(Grant Nos.DMS1501000 and DMS-1955249)。
文摘In this paper,we study large m asymptotics of the l 1 minimal m-partition problem for the Dirichlet eigenvalue.For any smooth domainΩ⊂R^(n)such that|Ω|=1,we prove that the limit lim_(m→∞)l^(1)_(m)(Ω)=c 0 exists,and the constant c 0 is independent of the shape ofΩ.Here,l^(1)_(m)(Ω)denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any m-partition ofΩ.
基金supported by the National Natural Science Foundation of China(Grant Nos.11631011,11626251)the China Postdoctoral Science Foundation(Grant No.2020M672398).
文摘The aim of this review is to introduce some recent results in eigenvalues problems for a class of degenerate elliptic operators with non-smooth coefficients,we present the explicit estimates of the lower bound and upper bound for its Dirichlet eigenvalues.
基金This research was supported by the National Natural Science Foundation of Chinathe Scientific Research Foundation of the Ministry of Education of China (02JA790014)+1 种基金the Natural Science Foundation of Fujian Province Education Department(JB00078)the Developmental Foundation of Science and Technology of Fuzhou University (2004-XQ-16)
文摘This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω , u = 0, x ∈δΩ, where Ω belong to R^n is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T. Yau et al.
文摘Simultaneous measurements of position and momentum are considered in n dimensions. We find, that for a particle whose position is strictly localized in a compact domain (spatial uncertainty) with non-empty boundary, the standard deviation of its momentum is sharply bounded by , while is the first Dirichlet eigenvalue of the Laplacian on D.
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.