Riemann boundary value problems for some K-regular functions in clifford analysis

In this paper, we study the R_m (m > 0) Riemann boundary value problems for regular functions, harmonic functions and bi-harmonic functions with values in a universal clifford algebra C(V_n,n). By using Plemelj formula, we get the solutions of R_m (m > 0) Riemann boundary value problems for regular functions. Then transforming the Riemann boundary value problems for harmonic functions and bi-harmonic functions into the Riemann boundary value problems for regular functions, we obtain the solutions of R_m (m > 0) Riemann boundary value problems for harmonic functions and bi-harmonic functions.     

Bi-f-harmonic map equations on singly warped product manifolds

Bi-f-harmonic maps are the critical points of bi-f-energy functional. This class of maps tends to integrate bi-harmonic maps and f-harmonic maps. In this paper, we show that bi-f-harmonic maps are not only an extension of f-harmonic maps but also an extension of bi-harmonic maps, and that there should exist many examples of proper bi-f-harmonic maps.In order to find some concrete examples of proper bi-f-harmonic maps, we study the basic properties of bi-f-harmonic maps from two directions which are conformal maps between the same dimensional manifolds and some special maps from or into a warped product manifold.     

Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schrödinger equations

We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrödinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the Nehari manifold.Furthermore, we show that using the so-called fibering method and the Lusternik–Schnirel’man theory there exist infinitely many solutions, actually a countable family of critical points, for such a semilinear bi-harmonic Schrödinger system under study in this work.     

Numerical differentiation for the second order derivatives of functions of two variables

A regularized optimization problem for computing numerical differentiation for the second order derivatives of functions with two variables from noisy values at scattered points is discussed in this article. We prove the existence and uniqueness of the solution to this problem, provide a constructive scheme for the solution which is based on bi-harmonic Green's function and give a convergence estimate of the regularized solution to the exact solution for the problem under a simple choice of regularization parameter. The efficiency of the constructive scheme is shown by some numerical examples.     

ON A BI-HARMONIC EQUATION INVOLVING CRITICAL EXPONENT: EXISTENCE AND MULTIPLICITY RESULTS

In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions: △2 u = K(x)u p , u > 0 in Ω , △u = u = 0 on Ω , where Ω is a smooth domain in R n , n 5, and p + 1 = 2 n n 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.     

Spectral stability results for higher-order operators under perturbations of the domain

We analyze the spectral behavior of higher-order elliptic operators when the domain is perturbed. We provide general spectral stability results for Dirichlet and Neumann boundary conditions. Moreover, we study the bi-harmonic operator with the so-called intermediate boundary conditions. We give special attention to this last case and analyze its behavior when the boundary of the domain has some oscillatory behavior. We will show that there is a critical oscillatory behavior and that the limit problem depends on whether we are above, below or just sitting on this critical value.     

Bi-harmonically excited pendulum: shifted resonances and quenching the low frequency excitation

The effect of the harmonic high-frequency vibration of the suspension point on a mathematical pendulum is well known [1]. It can be described as stiffening/softening or stabilization/destabilization of the up-pointing and down-pointing equilibriums for pure vertical or pure horizontal excitation. If the excitation is tilted the effect of biasing appears in addition [2]. The effects of bi-harmonic excitation are more complex. Two of them are discussed in the recent paper. The first one occurs if one of the excitation frequencies is high and another one is low. Then the so called shifted or conjugated resonances can be observed [3]. The second one takes place if the two excitation frequencies are high but their difference is small (slowly modulated high-frequency excitation according to [4]). It is shown that in that case it is possible to choose the bi-harmonic excitation in order to quench the resonance of the pendulum without changing its basic frequency. The analysis is fulfilled both for the external and parametric excitations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

MRA contextual-recovery extension of smooth functions on manifolds

In a recent paper, the first author introduced an MRA (multi-resolution or multi-level approximation) approach to extend an earlier work of Chan and Shen on image inpainting, from isotropic diffusion to anisotropic diffusion and from bi-harmonic extension to multi-level lagged anisotropic diffusion extension. The objective of the present paper is to extend and generalize this work to nonstationary smooth function extension to meet the goal of inpainting missing image features, while matching the existing image content without apparent visual artifact. Our result is formulated as an MRA contextual-recovery extension for the completion of smooth functions on manifolds by deriving an error formula, from which sharp error estimates can be derived. A novel estimate for the biharmonic operator derived in this paper is a formulation of the error bound in terms the volume, as opposed to the diameter, of the image hole.     

A penalty method for a mixed nonlinear complementarity problem

We propose a power penalty method for a mixed nonlinear complementarity problem (MNCP) and show that the solution to the penalty equation converges to that of the MNCP exponentially as the penalty parameter approaches infinity, provided that the mapping involved in the MNCP is both continuous and ξ-monotone. Furthermore, a convergence theorem is established when the monotonicity assumption on the mapping is removed. To demonstrate the usefulness and the convergence rates of this method, we design a non-trivial test MNCP problem arising in shape-preserving bi-harmonic interpolation and apply our method to this test problem. The numerical results confirm our theoretical findings.     

Existence of constant mean curvature 2-Spheres in Riemannian 3-spheres

We prove the existence of branched immersed constant mean curvature (CMC) 2-spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3-sphere is positively curved. To achieve this, we develop a min-max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi-harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min-max values to gain energy upper bounds for min-max sequences for almost every choice of mean curvature, and a Morse index estimate to obtain another uniform energy bound required to reach the remaining constant mean curvatures in the presence of positive curvature.     

On f-bi-harmonic maps and bi-f-harmonic maps between Riemannian manifolds

Both bi-harmonic maps and f-harmonic maps have some nice physical motivation and applications.Motivated largely by f-tension field not involving Riemannian curvature tensor, we attempt to formalize some large objects so as to broaden the notions of f-tension field and bi-tension field. We introduce a very large generalization of harmonic maps called f-bi-harmonic maps as the critical points of f-bi-energy functional, and then derive the Euler-Lagrange equation of f-bi-energy functional given by the vanishing of f-bi-tension field.Subsequently, we study some properties of f-bi-harmonic maps between the same dimensional manifolds and give a non-trivial example. Furthermore, we also study the basic properties of f-bi-harmonic maps on a warped product manifold so that we could find some interesting and complicated examples.     

Solutions of Super-Linear Elliptic Equations and Their Morse Indices

Mathematical Notes - We investigate the degenerate bi-harmonic equation $$\Delta_{m}^2 u=f(x,u)\quad \text{in} \ \ \Omega, \qquad u = \Delta u = 0\quad \text{on}\ \ \partial\Omega,$$ with $$m\ge...     

双调和型抛物方程的Schauder估计

该文给出了双调和型抛物方程初值问题解的Schauder估计,并且在适当的空间中证明了解的存在性与惟一性．类似于二阶情形,首先通过Fourier变换得到一个形式解,然后再利用位势理论和逼近方法得到解的正则性、唯一性及存在性．该方法简单而易懂．     

Average densities of frequency bands of wavelets

A function is called a wavelet if its integral translations and dyadic dilations form an orthonormal basis for L ²(?). The support of the Fourier transform of a wavelet is called its frequency band. In this paper, we study the relation between diameters and measures of frequency bands of wavelets, precisely say, we study the ratio of the measure to the diameter. This reflects the average density of the frequency band of a wavelet. In particular, for multiresolution analysis (MRA) wavelets, we do further research. First, we discuss the relation between diameters and measures of frequency bands of scaling functions. Next, we discuss the relation between frequency bands of wavelets and the corresponding scaling functions. Finally, we give the precise estimate of the measure of frequency bands of wavelets. At the same time, we find that when the diameters of frequency bands tend to infinity, the average densities tend to zero.     

Ciarlet–Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation

The Ciarlet–Raviart mixed finite element approximation is constructed to solve the constrained optimal control problem governed by the first bi-harmonic equation. The optimality conditions consisting of the state and the co-state equations is derived. Also, the a priori error estimates are analyzed. In the analysis of the a priori error estimates, the improved convergent rate of the higher order than existed results is proved. Some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate.     

Linear Approximation and Reproduction of Polynomials by Wilson Bases

Wilson bases are constituted by trigonometric functions multiplied by translates of a window function with good time frequency localization. In this article we investigate the approximation of functions from Sobolev spaces by partial sums of the Wilson basis expansion. In particular, we show that the approximation can be improved if polynomials are reproduced. We give examples of Wilson bases, which reproduce linear functions with the lowest-frequency term only.     

The growth of H-harmonic functions on the Heisenberg group

We discuss the relationship between the frequency and the growth of H-harmonic functions on the Heisenberg group.Precisely,we prove that an H-harmonic function must be a polynomial if its frequency is globally bounded.Moreover,we show that a class of H-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.     

Exact solution of the cubic-quintic Duffing oscillator

In this paper, we derive the exact solution of the cubic-quintic Duffing oscillator based on the use of Jacobi elliptic functions. We also showed that the exact angular frequency of this cubic-quintic Duffing equation is given in terms of the complete elliptic integral of the first kind.     

Hamilton-Jacobi方程的小波Galerkin方法

本文选择Daubechies小波尺度函数空间作为Galerkin方法的测试函数空间,并将其应用于Hamilton-Jacobi方程,得到了求解Hamilton-Jacobi方程的小波Galerkin方法的数值格式．由于小波在时间和频率上的局部性,本方法适用于处理具有奇异解的问题,可以有效地防止数值振荡．数值试验显示,本方法是有效的．     

Nonlinear approximation theory on compact groups

In this article we extend to the setting of band-limited functions on compact groups previous results bounding from below the percentage of energy, contained in the low frequency portion of the spectrum of a positive function defined on a cyclic group. Connections to signal recovery for positive functions, as well as partial spectral analysis, are also discussed.