A simple approach for determining the eigenvalues of the fourth-order Sturm–Liouville problem with variable coefficients

In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.     

HYPOELLIPTICITY OF NONSINGULAR CLOSED 1-FORMS ON COMPACT MANIFOLDS

ABSTRACT

Hypoellipticity of nonsingular closed 1-forms on compact manifolds is characterized by a diophantine condition on the generators of the group of periods of the form.     

Sturm-Liouville边值问题的正解存在性

该文的目的是研究Ｓｔｕｒｍ Ｌｉｏｕｖｉｌｌｅ边值问题的正解存在性．通过考察非线性项的局部特征获得了若干新的存在性结论．     

On stable quasi-harmonic maps and Liouville type theorems

We consider Liouville type problems of stable quasi-harmonic maps, by ``stable' we mean that the second variation of quasi-energy functional is nonnegative, and we prove that the stable quasi-harmonic maps must be constant under some geometry conditions.

     

关于椭圆型方程解的Liouville定理

在ｂ（ｘ）的较弱条件下，对下面的椭圆型方程，证整解的Ｌｉｏｕｖｉｌｌｅ 定理成立。     

Weyl symmetry and the Liouville theory

The flat-space conformal invariance and the curved-space Weyl invariance are simply related in dimensions greater than two. In two dimensions, the Liouville theory presents an exceptional situation, which we examine here. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 80–88, July, 2006.     

The Liouville equation in an annulus

We give the solutions to the Liouville equation in an annulus A of R² that satisfy a certain Neumann condition on each component of ∂A. As a consequence, we classify all the metrics of constant curvature in A that have constant geodesic curvature on ∂A.     

Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula

For a family of compact Riemann surfaces X_t of genus g > 1, parameterized by the Schottky space we define a natural basis of which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n  =  1. We introduce a holomorphic function F(n) on which generalizes the classical product for n  =  1 and g  =  1. We prove the holomorphic factorization formula

Notes on the Incompressible Euler and Related Equations on $\BR^N$\

The author reviews briefly some of the recent results on the blow-up problem for the incompressible Euler equations on R^N and also presents Liouville type theorems for the incompressible and compressible fluid equations.     

THE l~1-STABILITY OF A HAMILTONIAN-PRESERVING SCHEME FOR THE LIOUVILLE EQUATION WITH DISCONTINUOUS POTENTIALS

We study the l^1-stability of a Haxniltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 （2005）, 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the l^1-norm under a hyperbolic CFL condition which is in consistent with the l^1-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 （2008）, 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become l^1-unstable.