A Dirichlet problem for polyharmonic functions

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given. Project supported by NNSF of China.     

Polyharmonic dirichlet problems

A certain Dirichlet problem for the inhomogeneous polyharmonic equation is explicitly solved in the unit disc of the complex plane. The solution is obtained by modifying the related Cauchy-Pompeiu representation with the help of a polyharmonic Green function. Dedicated to Prof. S.M. Nikol’skii on the occasion of his 100th birthday and to the memory of P.G.L. Dirichlet on the occasion of his 200th birthday     

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on in Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed and will play an important role in the framework of circulant (2×2) matrix functions. Under this setting we will present the half Dirichlet problem for circulant (2×2) matrix functions on the unit ball of even dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solution to the Dirichlet problem on the unit ball for circulant (2×2) matrix functions and the solution to the classical Dirichlet problem as the special case, derive a decomposition of the Poisson kernel for matrix Laplace operator, and further obtain the decomposition theorems of solution space to the Dirichlet problem for circulant (2×2) matrix functions.     

On a new method for constructing the Green function of the Dirichlet problem for the polyharmonic equation

By using a special decomposition of the fundamental solution, we construct a new representation of the Green function of the Dirichlet problem for the polyharmonic equation in a ball.     

Construction of polynomial solutions to the Dirichlet problem for the polyharmonic equation in a ball

An algorithm is proposed for the analytical construction of a polynomial solution to Dirichlet problem for an inhomogeneous polyharmonic equation with a polynomial right-hand side and polynomial boundary data in the unit ball.     

On the solvability of some boundary value problems for the inhomogeneous polyharmonic equation with boundary operators of the Hadamard type

We study the properties of fractional integro-differential operators. As an application, we analyze the solvability of some boundary value problems for the inhomogeneous polyharmonic equation in the unit ball. These problems generalize the Dirichlet and Neumann problems to the case of fractional boundary operators.     

Minimal area problems for functions with integral representation

We study the minimization problem for the Dirichlet integral in some standard classes of analytic functions. In particular, we solve the minimal areaa ₂-problem for convex functions and for typically real functions. The latter gives a new solution to the minimal areaa ₂-problem for the classS of normalized univalent functions in the unit disc. Supported by NSF grant DMS-0412908.     

Singularities of the solution of the laplacian in domains with circular edges

In this paper we generalize the results from [4] to special domains with curved edges. For general elliptic boundary value problems the behavior of the solutions near arbitrary, smooth edges is analyzed by Maz'ja and Rossmann [3]. First following Dauge [1] we derive a regularity theorem for the solution of the Dirichlet problem of the Laplacian with a decomposition into edge singularities of nontensor product form. In this case the regularity of the remainder term in the decomposition corresponds to the one in the two-dimensional case [2]. Following [4] we obtain a refined decomposition where all singularity terms are of tensor product form. We illustrate our results with several examples.     

Malmheden's theorem revisited

In 1934 Malmheden [16] discovered an elegant geometric algorithm for solving the Dirichlet problem in a ball. Although his result was rediscovered independently by Duffin (1957) [8] 23 years later, it still does not seem to be widely known. In this paper we return to Malmheden's theorem, give an alternative proof of the result that allows generalization to polyharmonic functions and, also, discuss applications of his theorem to geometric properties of harmonic measures in balls in .     

A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem

According to an observation of A.V. Bitsadze from 1948 the Dirichlet problem for bianalytic functions is ill-posed. A natural boundary condition for the polyanalytic operator, however, is the Schwarz condition. An integral representation for the solutions in the unit disc to the inhomogeneous polyanalytic equation satisfying Schwarz boundary conditions is known. This representation is extended here to any simply connected plane domain having a harmonic Green function. Some other boundary value problems are investigated with some Dirichlet and Neumann conditions illuminating that just the Schwarz problem is a natural boundary condition for the Bitsadze operator.     

Two-Dimensional Multiple-Valued Dirichlet Minimizing Functions

We give some remarks on two-dimensional multiple-valued Dirichlet minimizing functions, including frequency, classification of branch points and their connections. As an application, we prove that blowing-up functions of a two-dimensional multiple-valued Dirichlet minimizing function are unique. This article is concluded with a boundary regularity theorem for two-dimensional multiple-valued Dirichlet minimizing functions.     

热弹性理论准静态问题解的一般公式及其应用

本文利用分解定理［１］求得热弹性理论的准静态问题在无限空间中的Ｇｒｅｅｎ函数组．从而，借助于文［２］中的互易定理建立了该问题解的一般公式．     

Nonlocal problems for some partial differential equations

Abstract Nonlocal problems for polyharmonic functions and for a special third order system in a half plane are studied which have applications in elasticity. Some of them are solved explicitly on the basis of solutions to related classical problems, others are reduced to the RlEMANN problem for serveral holomorphic functions.     

Normality and boundary behaviour of arbitrary and meromorphic functions along simple curves and applications

In this paper, we establish some theorems giving necessary and sufficient conditions for an arbitrary function defined in the unit disc of the complex plane to have boundary values along classes of an equivalence relation over simple curves. Our results generalize the well-known theorems on asymptotic and angular boundary behaviours of meromorphic functions (Lindelölf-, Lehto–Virtanen- and Seidel–Walsh-type theorems). The obtained results are applied to the study of boundary behaviour of meromorphic functions along curves using P-sequences, as well as in the proof of the uniqueness theorem similar to ?aginjan’s one. The constructed examples of functions show that the results cannot be improved.     

Variational problems in Clifford analysis

Using decomposition results for Sobolev spaces of Clifford‐valued functions into direct sums of subspaces of monogenic and co‐monogenic functions variational problems will be studied.These variational problems are equivalent to PD‐models by the choice of special operators of conboundary differentiation. By a Galerkin scheme we construct the monogenic part as a weak solution of a non‐linear problem. The co‐monogenic potential is the solution of a weak Dirichlet problem. Copyright © 2002 John Wiley & Sons, Ltd.     

二阶非线性椭圆偏微分方程Dirichlet问题的粘性解

研究了一类二阶非线性椭圆偏微分方程Dirichlet问题粘性解的存在性与唯一性。首先建立粘性解的比较定理，确保了解的唯一性，然后运用Perron方法构造出解。从而解决了这类问题的粘性解的存在性与唯一性。     

On duality theory in multiobjective programming

In this paper, we study different vector-valued Lagrangian functions and we develop a duality theory based upon these functions for nonlinear multiobjective programming problems. The saddle-point theorem and the duality theorem are derived for these problems under appropriate convexity assumptions. We also give some relationships between multiobjective optimizations and scalarized problems. A duality theory obtained by using the concept of vector-valued conjugate functions is discussed.The author is grateful to the reviewer for many valuable comments and helpful suggestions.     

单位复超球上多复变解析函数的边值问题

讨论了二元复变解析函数在单位复超球区域上的某些边值问题,包括Dirichlet问题和Riemann-Hilbert问题,利用Cauchy公式、Plemelj公式以及级数展开的方法,我们对不同标数的情形,给出了所提问题可解的充分必要条件.     

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.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem. Accepted 21 July 2001. Online publication 21 December 2001.