Decompositions of functions and Dirichlet problems in the unit disc

In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions.     

ON RIEMANN BOUNDARY VALUE PROBLEM FOR POLYANALYTIC FUNCTIONS ON THE REAL AXIS

In this article, Riemann boundary value problem with different factors for polyanalytic functions on the real axis is studied. The expression of solution and sufficient and necessary condition for solvability of the non-homogeneous Riemann boundary value problem are obtained.     

Nonexistence of local minima of supersolutions for the polyharmonic problems

In this note we study the nonexistence of local minima of the supersolutions of the polyharmonic equations on the balls under generalized homogeneous Dirichlet boundary conditions. Under suitable restriction on the dimensions, this means that a generalized clamped circular plate, which is pushed from below, cannot bend downwards even locally. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

双圆柱区域上的高阶四元数函数的Dirichlet边值问题

本文讨论了四元数分析中多圆柱区域上方程∂_zⁿu(z) = f 的Dirichlet 边值问题. 获得问题的可解条件, 并当可解条件满足时, 给出了解的积分表示.     

A note on polyharmonic functions

In this note we prove the following theorem:Suppose 0<p<∞ and α>−1. Then there is a constant C=C(p,m,n,α) such that

     

On Boundary Value Problems of Polyanalytic Functions on the Real Axis

In this article, we discuss the Riemann boundary value problems and the Hilbert boundary value problems of polyanalytic functions on the real axis, and both the explicit expressions of solutions and the conditions of solvability are obtained.     

Representation for the Green’s function of the Dirichlet problem for polyharmonic equations in a ball

We explicitly construct the Green’s function for the Dirichlet problem for polyharmonic equations in a ball in a space of arbitrary dimension. The formulas for the Green’s function are of interest in their own right. In particular, the explicit representations for a solution to the Dirichlet problem for the biharmonic equation are important in elasticity.     

The weak Dirichlet problem for Baire functions

Let be a simplex and a compact subset of the set of all extreme points of . We show that any bounded function of Baire class on can be extended to a function of affine class on . Moreover, can be chosen in such a way that .

     

Mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions

In this article, we reconsider the mixed boundary value problem on the unit circle for a pair of metaanalytic and analytic functions as in Du and Wang (2008) [9]. By adopting appropriate transformations, we convert the problem into two independent boundary value problems for analytic functions. We then obtain expressions of solution and condition of solvability for the mixed boundary value problem. The forms of the solutions and the condition of solvability here are rather dissimilar to those in Du and Wang (2008) [9]. But the equivalence is established at the end of this article.     

Expansions of Dirichlet to Neumann operators under perturbations

We obtain an asymptotic expansion of the Dirichlet to Neumann operator (DNO) for the Dirichlet problem on perturbations of the unit disk. We write our result in terms of pseudodifferential operators which themselves have expansions in the perturbation parameter. For a given power of the perturbation parameter, m > 0, and a given order, n < 0, we give an algorithm which allows for the expansion of the symbol of the DNO up to mth power in the perturbation parameter, with error terms belonging to symbols of order n.     

A mean value theorem on Dirichlet series

This paper deals with mean-value for the square of certain functionF(s) which has some characteristic properties of the Riemann zeta-function and its powers.Work supported by the University of the Basque Country.     

A critical elliptic problem for polyharmonic operators

In this paper, we study the existence of solutions for a critical elliptic problem for polyharmonic operators. We prove the existence result in some general domain by minimizing on some infinite-dimensional Finsler manifold for some suitable perturbation of the critical nonlinearity when the dimension of domain is larger than critical one. For the critical dimensions, we prove also the existence of solutions in domains perforated with the small holes. Some unstable solutions are obtained at higher level sets by Coron's topological method, provided that the minimizing solution does not exist.     

广义带形区域中的Dirichlet 问题

本文给出广义带形区域中Dirichlet 问题解的积分表示. 如果一类函数在广义带型区域内部调和并在边界上取值为零, 本文给出其需要满足的充要条件.     

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.     

一类Dirichlet边值逆问题

给出解析函数的一类Dirichlet边值逆问题的数学提法.依据解析函数Dirichlet边值问题和广义Dirichlet边值问题的理论,讨论了此边值逆问题的可解性.利用解析函数Dirichlet边值问题的Schwarz公式,给出了该边值逆问题的可解条件和解的表示式.     

Gaussian extended cubature formulae for polyharmonic functions

The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses values of linear functionals (integrals over hyperspheres) and is exact for all -harmonic functions, and consequently, for all algebraic polynomials of variables of degree .

     

A note on hyperharmonic and polyharmonic functions

We show that the spaces of harmonic functions with respect to the Poincaré metric in the unit ball B^N in have many different properties depending upon whether N is even or odd.