The Landau–Kolmogorov problem for the Laplace operator on a ball

In this paper we solve the problem of maximizing the value of the Laplace operator at the origin for functions such that the second degree of the Laplace operator belongs to the space L_∞ on the unit ball of the Euclidean space. The problem is solved under restrictions on the uniform norm of a function and the L_∞-norm of the second degree of the Laplace operator of this function.     

Eigenvalue inequalities for the clamped plate problem of the ■ operator

The ■ operator is introduced by Xin(2015), which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend the ■ operator to a more general elliptic differential operator ■, and investigate the clamped plate problem of the bi-■ operator,which is denoted by ■ on the complete Riemannian manifolds. A general formula of eigenvalues for the ■ operator is established. Applying this formula, we estimate the eigenvalues on th...     

Eigenvalue asymptotics for the Schrödinger operator with perturbed periodic potential

Summary We consider the Schrödinger operatorH=–+W+V acting inL ²( ^m ),m2, with periodic potentialW perturbed by a potentialV which decays slowly at infinity. We study the asymptotic behaviour of the discrete spectrum ofH near any given boundary point of the essential spectrum.Oblatum 1-VII-1991 & 20-I-1992Partly supported by the Bulgarian Science Foundation under contract No MM 8/1991     

Eigenvalue estimates for the Dirac operator on quaternionic K?hler manifolds

We consider the Dirac operator on compact quaternionic K?hler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space. Received April 21, 1998; in final form June 16, 1998     

Eigenvalue estimates for the Dirac operator on Kähler–Einstein manifolds of even complex dimension

In the case of a Kähler–Einstein manifold of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.     

Estimate from Below of the First Eigenvalue for the Beltrami–Laplace Operator in the Three-Dimensional Space

In this work the following theorem is proved by elementary methods. Theorem. For all congruence-subgroups of the group PSL₂ the following inequality holds: ₁ – , where is the ring of all Gaussian integers.     

Eigenvalue dynamics of a <Emphasis Type="Italic">PT</Emphasis>-symmetric Sturm–Liouville operator and criteria for similarity to a self-adjoint or a normal operator

The dynamics of the eigenvalues of a family of Sturm–Liouville operators with complex integrable PT-symmetric potential on a finite interval is studied. In the model case of the complex Airy operator, a criterion for the similarity of operators in the family to self-adjoint and normal operators is stated and the exceptional parameter values corresponding to multiple eigenvalues are analytically calculated.     

Laplace operator with δ-like potentials

We study the Laplace operator in a punctured domain in a Hilbert space. We obtain an analog of the Green formula and a class of self-adjoint extensions of the Laplacian. We also investigate a certain class of well-posed problems.     

Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold

We consider the eigenvalue problem of the Schrödinger operator with the magnetic field on a compact Riemannian manifold. First we discuss the least eigenvalue. We give a representation of the least eigenvalue by the variational formula and give a relation to the least eigenvalue of the Schrödinger operator without the magnetic field. Second, we discuss the asymptotic distribution of eigenvalues by obtaining the asymptotic expansion of the kernel of semigroup. Here we use the theory of asymptotic expansion for Wiener functionals.     

Finsler流形上Laplace比较定理及其应用

本文研究了Finsler流形上距离函数的Laplacian.利用Schwarz不等式和[5]中主要方法,获得了具有负曲率的Laplacian比较定理,进而得到了Finsler流形上第一特征值的下界估计.     

Finsler空间中的Laplace算子及其性质

在Finsler空间中给出了一种非线性的Laplace算子Δ,得到了Laplace算子Δ满足的性质,同时指出了Δ与Riemann空间中Laplace算子的异同.     

Green functions and eigenfunction expansions for the square of the Laplace–Beltrami operator on plane domains

For a certain class of domains Ω⊂ℂ with smooth boundary and Δtilde;_Ω=w ²Δ the Laplace–Beltrami operator with respect to the Poincaré metric ds ²=w(z)^-2 dz dz on Ω, we (1) show that the Green function for the biharmonic operator Δtilde;_Ω ², with Dirichlet boundary data, is positive on Ω×Ω; and (2) obtain an eigenfunction expansion for the operator Δtilde;_Ω, which reduces to the ordinary non-Euclidean Fourier transform of Helgason for Ω=𝔻 (the unit disc). In both cases the proofs go via uniformization, and in (1) we obtain a Myrberg-like formula for the corresponding Green function. Finally, the latter formula as well as the eigenfunction expansion are worked out more explicitly in the simplest case of Ω an annulus, and a result is established concerning the convergence of the series ∑ _ω∈G (1-|ω0|²) ^s for G the covering group of the uniformization map of Ω and 0<s<1. Received: August 21, 2000?Published online: October 30, 2002 RID="*" ID="*"The first author was supported by GA AV CR grants no. A1019701 and A1019005.     

Non-stretch mappings for a sharp estimate of the Beurling–Ahlfors operator

In this paper we identify certain classes of non-stretch mappings that enjoy a sharp estimate of the Beurling–Ahlfors operator. We first make use of a property of subharmonic functions to prove that the Bañuelos–Wang conjecture and the Iwaniec conjecture are true for a class of mappings that satisfy a quasilinear conjugate Beltrami equation. By utilizing the principal solutions of Beltrami equations, we further explicitly construct some classes of non-stretch mappings for which the Bañuelos–Wang conjecture and the Iwaniec conjecture are true.     

Finsler流形上的Laplace算子

该文对Finsler流形上的微分式定义了整体内积，进而引入δ算子和Laplace算子。该文还给出了δ算子的局部坐标表达式并且证明了Laplace算子可以看成是Riemann流形上Laplace算子在Finsler流形上的扩张。     

Lower Bound of the First Eigenvalue for the Laplace Operator on Compact Riemannian Manifold

Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2~(1/m-1),2~(1/2)}, Such thatλ_1≥π~2/d~2·1/(2-(11)/(2π~2))+11/2π~2e~cm、(?)     

A remark on the Gaussian lower bound for the Neumann heat kernel of the Laplace–Beltrami operator

We adapt in the present note the perturbation method introduced in Choulli and Kayser (Positivity 19(3):625–646, 2015) to get a lower Gaussian bound for the Neumann heat kernel of the Laplace–Beltrami operator on an open subset of a compact Riemannian manifold.