LOWER BOUNDS OF DIRICHLET EIGENVALUES FOR A CLASS OF FINITELY DEGENERATE GRUSHIN TYPE ELLIPTIC OPERATORS

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.

Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Let λkbe the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator-ΔB defined on a stretched cone B0 ■ [0,1) × X with boundary on {x1 = 0}. More precisely,ΔB=(x1αx1)2+ α2x2+ + α2xnis also called the cone Laplacian. In this paper,by using Mellin-Fourier transform,we prove thatλk Cnk2 n for any k 1,where Cn=(nn+2)(2π)2(|B0|Bn)-2n,which gives the lower bounds of the Dirchlet eigenvalues of-ΔB. On the other hand,by using the Rayleigh-Ritz inequality,we deduce the upper bounds ofλk,i.e.,λk+1 1 +4n k2/nλ1. Combining the lower and upper bounds of λk,we can easily obtain the lower bound for the first Dirichlet eigenvalue λ1 Cn(1 +4n)-12n2.

A Probabilistic Approach to the First Dirichlet Eigenvalue on Non-compact Riemannian Manifold

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.

Estimates of Dirichlet Eigenvalues for One-Dimensional Fractal Drums

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.

Large m asymptotics for minimal partitions of the Dirichlet eigenvalue

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.

Dirichlet Eigenvalue Ratios for the p-sub-Laplacian in the Carnot Group

We prove some new Hardy type inequalities on the bounded domain with smooth boundary in the Carnot group. Several estimates of the first and second Dirich- let eigenvalues for the p-sub-Laplacian are established.

Dirichlet Eigenvalue Problem of Degenerate Elliptic Operators with Non-Smooth Coefficients

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.

THE EIGENVALUE PROBLEM FOR THE LAPLACIAN EQUATIONS

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.

The Uncertainty Principle in Terms of Isoperimetric Inequalities

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.

New point view of spectral gap in functional spaces for birth-death processes

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.