ON THE FIRST EIGENVALUE OF THE MEAN FINSLER-LAPLACIAN

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.

Sharp Bounds for the First Eigenvalue of Symmetric Markov Processes and Their Applications

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.

Estimates and Monotonicity of the First Eigenvalues Under the Ricci Flow on Closed Surfaces

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.

First Eigenvalue Monotonicity for the p-Laplace Operator under the Ricci Flow

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）]

POSITIVE SOLUTIONS FOR PARAMETRIC EQUIDIFFUSIVE p-LAPLACIAN EQUATIONS

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.

POSITIVE SOLUTIONS TO FOURTH-ORDER NEUMANN BOUNDARY VALUE PROBLEM

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.

Blow-up for Doubly Degenerate Nonlinear Parabolic Equations

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.

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.

EXISTENCE OF POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM FOR ONE-DIMENSIONAL p-LAPLACIAN

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.

MULTIPLE POSITIVE PERIODIC SOLUTIONS TO SINGULAR DIFFERENTIAL EQUATION WITH STATE-DEPENDENT DELAY

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.