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 Dirichl...
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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. (C) 2009 Elsevier Inc. All rights reserved.
We consider a homogeneous polyanalytic differential equation of order n in a simply-connected domain D with a smooth boundary partial derivativeD in the complex plane C. We pose and then prove solvability of a general...
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We consider a homogeneous polyanalytic differential equation of order n in a simply-connected domain D with a smooth boundary partial derivativeD in the complex plane C. We pose and then prove solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is done by reducing the problem into n classical Riemann problems of linear conjugation for holomorphic functions, the solution of which is available in the literature.
Iterating the Pompeiu integral operator leads to area integral operators being right inverses to proper powers of the Cauchy-Riemann differential operator. These integral operators serve to solve boundary value proble...
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Iterating the Pompeiu integral operator leads to area integral operators being right inverses to proper powers of the Cauchy-Riemann differential operator. These integral operators serve to solve boundary value problems for differential equations the leading term of which is some power of the Cauchy-Riemann operator. This situation in investigated also for products of powers of the Cauchy-Riemann operator and its complex conjugate. Besides the complex cases of one and of several complex variables the consideration can be extended to the'hyper"complex and actually is extended to the Clifford analysis case. The generalized Cauchy-Pompeiu formula is used to orthogonally decompose the set L-2(D,C) with respect to polyanalytic and polyharmonic functions.
We consider a non-homogeneous polyanalytic partial differential equation of order n in a simply-connected domain D with smooth boundary partial derivativeD in the complex plane C. Initially we transform the given equa...
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We consider a non-homogeneous polyanalytic partial differential equation of order n in a simply-connected domain D with smooth boundary partial derivativeD in the complex plane C. Initially we transform the given equation into an equivalent system of integro-differential equations and then find the general solution of the former in W-n,W-p(D). Next we pose and prove the solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is effected by first reducing the Riemann problem to a corresponding one for a polyanalytic function. The latter is solved by first transforming it into n classical Riemann problems of linear conjugation for n holomorphic functions expressed in terms of the analytic functions which define the polyanalytic function. The solution of the classical Riemann problem is available in the literature.
Let Omega subset of R-N ( N greater than or equal to 2) be an unbounded domain, and L-m be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other thi...
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Let Omega subset of R-N ( N greater than or equal to 2) be an unbounded domain, and L-m be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing L-1-solutions to L-m (in Omega) approximate all L-1-solutions to L-m (in Omega), provided there exist real numbers R-j --> infinity, epsilon greater than or equal to 0, and it sequence {y(j)} such that B(y(j), epsilon) boolean AND Omega = circle divide and \A(y(j), R-j, R-N\Omega)\/R-j(N) > epsilon For All j, where \.\ means the volume and [GRAPHICS] for z is an element of R-N, R > 0 and D subset of R-N. For m = 2, we can replace the volume density by the capacity-density. It appears that the problem is related to this characterization of largest sets on which a nonzero polynomial solution to L-m may vanish, along with its (m-1)-derivarives. We also study a similar approximation problem for polyanalytic functions in C. (C) 2000 Academic Press.
Given is a nonlinear non-homogeneous polyanalytic differential equation of order n in a simply-connected domain D in the complex plane. Initially we prove (under certain conditions) the existence of its general soluti...
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Given is a nonlinear non-homogeneous polyanalytic differential equation of order n in a simply-connected domain D in the complex plane. Initially we prove (under certain conditions) the existence of its general solution in W-n,(p)(D) by first transforming it into a system of integro-differential equations. Next we prove the solvability of a generalized Riemann-Hilbert problem for the differential equation. This is effected by first reducing the boundary value problem posed to a corresponding one for a polyanalytic function. The latter is then transformed into n classical Riemann-Hilbert problems for holomorphic functions, whose solutions are known in the literature:
Using methods of the theory of generalized analytic functions, the present paper investigates the boundary behaviour of non-holomorphic functions (such as polyanalytic ones) provided the areolar derivative ∂/∂zˉ belo...
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Using methods of the theory of generalized analytic functions, the present paper investigates the boundary behaviour of non-holomorphic functions (such as polyanalytic ones) provided the areolar derivative ∂/∂zˉ belong to an Lp space,
Boundary properties
polyanalytic functions
inhomogeneous Cauchy-Riemann equation
cluster sets
Abstract: Let $f(z) = \sum \nolimits _{k = 0}^n {{{\bar z}^k}{f_k}(z)}$ where the functions ${f_0},{f_1}, \cdots ,{f_n}$ are analytic on some annular neighborhood A of the point $\infty$ and ${f_n} \equiv 1$ o...
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Abstract: Let $f(z) = \sum \nolimits _{k = 0}^n {{{\bar z}^k}{f_k}(z)}$ where the functions ${f_0},{f_1}, \cdots ,{f_n}$ are analytic on some annular neighborhood A of the point $\infty$ and ${f_n} \equiv 1$ on A and z denotes the complex conjugate of z. If f does not vanish on A it is shown that the functions ${f_0},{f_1}, \cdots ,{f_{n - 1}}$ have a nonessential isolated singularity at the point infinity.
Abstract: Let $f(z,\bar z)$ be a bianalytic function which omits the value zero in some deleted neighborhood $A$ of an isolated singularity ${z_0}$. It is shown that there is a function $g(z)$ analytic on $A$ ...
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Abstract: Let $f(z,\bar z)$ be a bianalytic function which omits the value zero in some deleted neighborhood $A$ of an isolated singularity ${z_0}$. It is shown that there is a function $g(z)$ analytic on $A$ and a function $h(z,\bar z)$ bianalytic on $A$ with a nonessential singularity at ${z_0}$ such that $f(z,\bar z) = g(z)h(z,\bar z)$ on $A$.
Let f, g, and h be polyanalytic in an annular neighborhood A of a complex number z0, finite or infinite, such that g and h do not have an essential singularity at z0 and g - h is not identically zero on A. It is shown...
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