Jun Ling
We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.
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Estimate for the first Dirichlet eigenvalue of $p-$Laplacian on non-compact manifolds
本文研究了非紧黎曼流形上 $p$-拉普拉斯算子第一个狄利克雷特征值的估计问题，得出了具有非负里奇曲率的完整非紧流形上的精确下界结果。
New inequalities for eigenvalues of the Dirichlet Laplacian on the hyperbolic space
本文基于对狄利克雷拉普拉斯算子特征值通用不等式的研究，证明了超几何空间上相关特征值的新不等式，并验证了陈猜想在两种特殊有界域中的成立，允许ε损失。
We give a new estimate on the lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature. The result improves the previous estimates.
In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice $\Z^n.$ This partially answers a question posed by Chung and Oden.
In this paper, the third of its series, we prove that the sobolev spaces of L^p_k approximate solutions to the Dirichlet problem for the epsilon-Yang Mills equations on a four dimensional disk, carry a natural manifold structure (more precisely a natural structure of Banach bundle), for p(k+1)> 4. All results apply also if the four-dimensional disk is replaced by a general compact manifold with boundary, and SU(2) is replaced by any compact Lie group. We also construct bases for the tangent space to the space of approximate solutions, thus showing that this space is 8-dimensional for epsilon s…
The purpose of this paper is to investigate the parabolic deformed Hermitian-Yang-Mills equation with hypercritical phase in a smooth domain $Ω\subset \mathbb{C}^{n}$. By using $J$-functional, we are able to prove the convergence of solutions. As an application, we give an alternative proof of the Dirichlet problem for deformed Hermitian-Yang-Mills equation.
Some universal inequalities for Dirichlet eigenvalues of the Laplacian on a Euclidean convex domain
本文建立了欧几里得凸域上拉普拉斯算子狄利克雷本征值的两个通用不等式，探讨了相关数学性质。
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct continuous solutions of the associated Dirichlet problem.
In this article we will explore Dirichlet Laplace eigenvalues on balls on spherically symmetric manifolds. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radius. As a special case we will show that the Dirichlet Laplace eigenvalues on balls with small radius on the sphere are smaller than the corresponding eigenvalues on the Euclidean ball with the same radius. While the opposite is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces.
In this paper we complete the proof of the existence of multiple solutions (and, in particular, non minimal ones), to the epsilon-Dirichlet problem obtained as a variational problem for the SU(2)-epsilon-Yang Mills functional. This is equivalent to proving the existence of multiple solutions to the Dirichlet problem for the SU(2)-Yang Mills functional with small boundary data. In the first paper of this series this non-compact variational problem is transformed into the finite dimensional problem of finding the critical points of the function J(q), which is essentially the Yang Mills functiona…
In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minimal submanifolds in Euclidean space. Lastly we give a sharp estimate of the eigenvalue (with outer Dirichlet and inner Neumann boundaries) in terms of the Dircihlet eigenvalue, and also we give an upper bound of the eigenvalue with inner Dirichlet and outer Neumann problems by the…
The first Dirichlet eigenvalue and the width
该研究针对非负Ricci曲率、平均凸边界的测地球，提出定量显式不等式，用第一Dirichlet特征值与其精确下界的谱间隙界定测地球宽度，拓展了相关谱几何结论。
Several isoperimetric inequalities of Dirichlet and Neumann eigenvalues of the Witten-Laplacian
本文利用重排技术与试验函数构造，在固定加权体积条件下建立空间形式中有界区域上 Witten-Laplacian 多个 Dirichlet 与 Neumann 特征值的等周不等式，推广了 Laplacian 的经典结果。
Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds
本文研究了共形平坦黎曼流形上Dirichlet Laplacian的特征值问题，并在双曲空间$\mathbb{H}^n(-1)$上建立了特征值的普遍不等式。
The Dirichlet eigenvalue problems for some concave elliptic Hessian operators
本文研究一类广泛凹椭圆Hessian算子的Dirichlet特征值问题，在温和约束下证明了首个非零特征值及特征函数的存在性与正则性，并推广至Gårding-Dirichlet算子，建立了分岔理论。
Asymptotic behavior of the first Dirichlet eigenvalue of AHE manifolds
本文研究AHE流形测地球第一Dirichlet特征值随半径趋于无穷的递减速率，证明共形无穷远为非负Yamabe型时其两项渐近展开与双曲空间相同。
In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.
We prove a sharp Zhong-Yang type eigenvalue lower bound for closed Riemannian manifolds with control on integral Ricci curvature.
Dirichlet eigenvalues of Schwarz triangle (3/2 3/2 3/2)
本文计算球面上等边Schwarz三角形(3/2 3/2 3/2)的前两个Dirichlet特征值及特征函数。
Estimates for Eigenvalues of the Dirichlet Laplacian on Riemannian Manifolds
本文重新探讨了完备黎曼流形有界域上 Dirichlet Laplacian 的特征值问题，基于 Li-Yau 和 Yang 不等式等经典结果推导出特征值上下界，并对射影空间及其极小子流形给出了特征值下界的显式估计。
In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-Wang\cite{HHW} and Hassannezhad-Miclo \cite{HM}. Then, some Raulot-Savo-type estimates on the eigenvalues of the DtN maps introduced are derived.
Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture that this number is controlled by the isoperimetric ratio of the domain. This has applications to the nodal deficiency of eigenfunctions and is closely related to a long-standing conjecture of Yau on the Hausdorff measure of nodal sets.
We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.
We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem
The paper by G. Liu [arxiv:2109.02561] contains an error. In this note, I give a brief review of the problem and indicate what the error is.
We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the models used are $CP^q$ or the complex hyperquadric, these bounds also give gap phenomena and comparison results.
We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball in these rotationally symmetric spaces, including the real space forms of constant curvature. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space contro…
Comparisons of Dirichlet, Neumann and Laplacian eigenvalues on graphs and their applications
本文研究图上Dirichlet、Neumann和Laplacian特征值之间的比较关系，讨论刚性情形，并给出Lichnerowicz型、Fiedler型和Friedman型估计等应用，推广了Hua-Huang-Wang及作者先前的工作。
Eigenvalue estimates for the poly-Laplace operator on lattice subgraphs
本文引入了具有 Dirichlet 边界条件的子图上的离散多重 Laplace 算子，推导了格图 $\mathbb{Z}^{d}$ 有限子图上其前 $k$ 个特征值之和的上下界，并证明了高阶特征值与低阶特征值平方之间的不等式关系。
We prove the long-standing Payne conjecture that the $k^{\text{th}}$ eigenvalue in the buckling problem for a clamped plate is not less than the ${k+1}^{\text{st}}$ eigenvalue for the membrane of the same shape which is fixed on the boundary. Moreover, we show that the Payne conjecture is still true for $n$-dimensional case ($n\ge 2)$.
Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary
研究通过构造(ε,ρ)近邻图的截断图拉普拉斯算子谱来近似流形上的狄利克雷特征值，证明在满足曲率、注入半径和直径约束的流形类中，特征值收敛性。
In this paper, we investigate the monotonicity of the first Steklov--Dirichlet eigenvalue on eccentric annuli with respect to the distance, namely $t$, between the centers of the inner and outer boundaries of an annulus. We first show the differentiability of the eigenvalue in $t$ and obtain an integral expression for the derivative value in two and higher dimensions. We then derive an upper bound of the eigenvalue for each $t$, in two dimensions, by the variational formulation. We also obtain a lower bound of the eigenvalue, given a restriction that the two boundaries of the annulus are suffi…
In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.
The sharp lower bound of the first Dirichlet eigenvalue for geodesic balls
本研究在非负Ricci曲率的完备非紧黎曼流形框架下，得到测地球第一Dirichlet特征值的精确下界，由此推导出测地球的精确Poincaré不等式，推进了Li-Schoen的相关结果。
In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \cite{ChengYang07}, the Yang-type inequality for eigenvalues of the weighted Hodge Laplacian are optimal in the sense of the order of eigenvalues.
We prove universal inequalities for Laplacian eigenvalues with Dirichlet boundary conditions on subsets of certain discrete groups. The study of universal inequalities on Riemannian manifolds was initiated by Weyl, Polya, Yau, and others. Here we focus on a version by Cheng and Yang. Specifically, we prove Yang-type universal inequalities for Cayley graphs of finitely generated amenable groups, as well as for the d-regular tree (simple random walk on the free group).
In this paper (Part I) and its sequels (Part II and Part III), we analyze the structure of the space of solutions to the epsilon-Dirichlet problem for the Yang-Mills equations on the 4-dimensional disk, for small values of the coupling constant epsilon. These are in one-to-one correspondence with solutions to the Dirichlet problem for the Yang Mills equations, for small boundary data. We prove the existence of multiple solutions, and, in particular, non minimal ones, and establish a Morse Theory for this non-compact variational problem. In part I, we describe the problem, state the main theore…
Prescription of finite Dirichlet eigenvalues and area on surface with boundary
本文证明在固定带边紧曲面上，可构造度量同时实现任意有限递增Dirichlet特征值序列和指定面积。
In this paper, we study a first Dirichlet eigenfunction of the weighted $p$-Laplacian on a bounded domain in a complete weighted Riemannian manifold. By constructing gradient estimates for a first eigenfunction, we obtain some relationships between weighted $p$-Laplacian first eigenvalues. As an immediate application, we also obtain some eigenvalue comparison results between the first Dirichlet eigenvalue of the weighted Laplacian, the first clamped plate eigenvalue and the first buckling eigenvalue.
In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus $γ$ and the area. The equality in Yang-Yau's estimate is attained for $γ=0$ by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for $γ=2$. In the present article we combine techniques from algebraic geometry and minimal surface theory to show that Yang-Yau's inequality is strict for all genera $γ> 2$. Previously this was only known for $γ=1$. In the se…
We try to present an estimate relating the first Dirichlet and Neumann eigenvalues of a compact bordered Riemannian surface.
We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considerin…
The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots . A natural toy model for further investigations is the Möbius strip, a non-orientable surface with Euler characteristic $0$, and particularly the "square" Möbius strip whose eigenvalues have higher multiplicities. In this case…
Conformally compact and higher conformal Yang-Mills equations
本文在共形紧致流形上研究Yang-Mills方程的边界问题，发现磁性Dirichlet数据的阻碍流，构造高阶共形边界算子，并给出重整化作用泛函，从而建立高阶共形Yang-Mills方程。
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