50 research outputs found
Eigenvalues of the Laplacian on Riemannian manifolds
For a bounded domain with a piecewise smooth boundary in a complete
Riemannian manifold , we study eigenvalues of the Dirichlet eigenvalue
problem of the Laplacian. By making use of a fact that eigenfunctions form an
orthonormal basis of in place of the Rayleigh-Ritz formula, we
obtain inequalities for eigenvalues of the Laplacian. In particular, for lower
order eigenvalues, our results extend the results of Chen and Cheng \cite{CC}.Comment: 17 page
Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations
Asymptotic stability and boundedness have been two of most popular topics in the study of stochastic functional differential equations (SFDEs) (see e.g. Appleby and Reynolds (2008), Appleby and Rodkina (2009), Basin and Rodkina (2008), Khasminskii (1980), Mao (1995), Mao (1997), Mao (2007), Rodkina and Basin (2007), Shu, Lam, and Xu (2009), Yang, Gao, Lam, and Shi (2009), Yuan and Lygeros (2005) and Yuan and Lygeros (2006)). In general, the existing results on asymptotic stability and boundedness of SFDEs require (i) the coefficients of the SFDEs obey the local Lipschitz condition and the linear growth condition; (ii) the diffusion operator of the SFDEs acting on a C2,1-function be bounded by a polynomial with the same order as the C2,1-function. However, there are many SFDEs which do not obey the linear growth condition. Moreover, for such highly nonlinear SFDEs, the diffusion operator acting on a C2,1-function is generally bounded by a polynomial with a higher order than the C2,1-function. Hence the existing criteria on stability and boundedness for SFDEs are not applicable andwesee the necessity to develop new criteria. Our main aim in this paper is to establish new criteria where the linear growth condition is no longer needed while the up-bound for the diffusion operator may take a much more general form
Bounds for eigenvalue ratios of the Laplacian
For a bounded domain with a piecewise smooth boundary in an
-dimensional Euclidean space , we study eigenvalues of the
Dirichlet eigenvalue problem of the Laplacian. First we give a general
inequality for eigenvalues of the Laplacian. As an application, we study lower
order eigenvalues of the Laplacian and derive the ratios of lower order
eigenvalues of the Laplacian.Comment: 14 page
Stochastic population dynamics under regime switching II
AbstractThis is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69β84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population
Estimates on the first two buckling eigenvalues on spherical domains
In this paper, we study the first two eigenvalues of the buckling problem on
spherical domains. We obtain an estimate on the second eigenvalue in terms of
the first eigenvalue, which improves one recent result obtained by Wang-Xia in
[7].Comment: This article has been submitted for publication on 2009-04-2