Harmonic Polynomials and Dirichlet-Type Problems

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on R N . The Dunkl operators are denoted by T j for 1 ≤ j ≤ N , and the Laplacian Δ κ = ∑ j = 1 N T j 2 . This paper finds the homogeneous harmonic polynomials annihilated by all T j for j > 2 . The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.

This chapter takes up several independent topics in Euclidean Fourier analysis, all having some bearing on the subject of partial differential equations. Section 1 elaborates on the relationship between the Fourier transform and the Schwartz space, the subspace of $L^{1}(\mathbb{R}^{N})$ consisting of smooth functions with the property that the product of any iterated partial derivative of the function with any polynomial is bounded. It is possible to make the Schwartz space into a metric space, and then one can consider the space of continuous linear functionals; these continuous linear functionals are called “tempered distributions.” The Fourier transform carries the space of tempered distributions in one-one fashion onto itself. Section 2 concerns weak derivatives, and the main result is Sobolev’s Theorem, which tells how to recover information about ordinary derivatives from information about weak derivatives. Weak derivatives are easy to manipulate, and Sobolev’s Theorem is therefore a helpful tool for handling derivatives without continually having to check the validity of interchanges of limits. Sections 3–4 concern harmonic functions, those functions on open sets in Euclidean space that are annihilated by the Laplacian. The main results of Section 3 are a characterization of harmonic functions in terms of a mean-value property, a reflection principle that allows the extension to all of Euclidean space of any harmonic function in a half space that vanishes at the boundary, and a result of Liouville that the only bounded harmonic functions in all of Euclidean space are the constants. The main result of Section 4 is a converse to properties of Poisson integrals for half spaces, showing that harmonic functions in a half space are given as Poisson integrals of functions or of finite complex measures if their $L^{p}$ norms over translates of the bounding Euclidean space are bounded. Sections 5–6 concern the Calderon–Zygmund Theorem, a far-reaching generalization of the theorem concerning the boundedness of the Hilbert transform. Section 5 gives the statement and proof, and two applications are the subject of Section 6. One of the applications is to Riesz transforms, and the other is to the Beltrami equation, whose solutions are “quasiconformal mappings.” Sections 7–8 concern multiple Fourier series for smooth periodic functions. The theory is established in Section 7, and an application to traces of integral operators is given in Section 8.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsShif BerhanuSHIF BERHANU graduated with a B.Sc. degree from Addis Ababa University, Ethiopia, in 1980. In 1987 he received his Ph.D. from Rutgers University under the direction of Francois Trevès. He is a professor of mathematics at Temple University. Berhanu's main area of research involves complex vector fields, that is, firstorder partial differential operators with complex-valued coefficients. Currently, he, P. Cordaro, and J. Hounie are collaborating to write a book on the subject of complex vector fields.Ahmed MohammedAHMED MOHAMMED received his B.Sc. in 1983 and M.Sc. in 1986 from Addis Ababa University, Ethiopia. He obtained his Ph.D. in 1995 from Temple University under the guidance of Professor Cristian E. Gutierrez. Between 1995 and 2000, he taught in the Department of Mathematics and Computer Science at Kuwait University, Kuwait. He held a visiting position at Temple University from 2000 to 2002. Currently he is an associate professor in the Department of Mathematical Sciences at Ball State University. If he is not doing mathematics in his office or teaching a class, chances are you'll find him at the Starbucks in the university book store.

A family of sense–preserving complex valued harmonic polynomials that are close-to-convex on the open unit disk is introduced. By taking limits as the degree of the polynomials tends to infinity, a famiiy of close-to-convex harmonic mappings is also obtained. Similar results are considered for mappings of the punctured disk.

BEM solution of Poisson's equation in two dimensions with polynomial interpolation of the source function

Shift operators on harmonic Hilbert function spaces on real balls and von Neumann inequality

In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic -like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas’yanyuk [10] and Brickman [3] for analytic functions, but with a simpler proof than theirs. A number of consequences of the characterization, and examples are also presented. Further investigation provides a new method to construct univalent harmonic mappings with the help of an improved distortion theorem.

Marden's theorem characterizes the critical points of complex polynomials of degree 3 in a nice geometrical way. Our proof of the theorem is based directly on the defining property of ellipses.

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$-harmonic functions on $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$-Poisson kernel $P_{\alpha, \beta} (z, \zeta)$. A characterization by hypergeometric functions of separately $(\alpha, \beta)$-harmonic functions which are also $m$-homogeneous is given, it is used to obtain series expansion of separately $(\alpha, \beta)$-harmonic functions. Basic $H^p$ theory of such functions is developed: integral representations by measures and $L^p$ functions on $\mathbb T^n$, norm and weak$^\ast$ convergence at the distinguished boundary $\mathbb T^n$. Weak $(1,1)$-type estimate for a restricted non-tangential maximal function $M_{A, B}^{\mathrm{NT}}$ is derived. We show that slice functions $u(z_1, …, z_k, \zeta_{k+1}, …, \zeta_n)$, where some of the variables are fixed, belong in the appropriate space of separately $(\alpha', \beta')$-harmonic functions of $k$ variables. We prove a Fatou type theorem on a. e. existence of restricted non-tangential limits for these functions and a corresponding result for unrestricted limit at a point in $\mathbb T^n$. Our results extend earlier results for $(\alpha, \beta)$-harmonic functions in the disc and for $n$-harmonic functions in $\mathbb D^n$.

In this paper, we consider $p$-harmonic functions on complete Riemannian manifolds and give different proofs of some main theorems by Chang-Chen-Wei, in [3]. Moreover, we are able to refine their results in case of weakly $p$-harmonic functions. Some applications to study the connectedness at infinity of stable minimal hypersurfaces are also given.

Harmonic polynomials, hyperspherical harmonics, and atomic spectra

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$-harmonic functions on $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$-Poisson kernel $P_{\alpha, \beta} (z, \zeta)$. A characterization by hypergeometric functions of separately $(\alpha, \beta)$-harmonic functions which are also $m$-homogeneous is given, it is used to obtain series expansion of separately $(\alpha, \beta)$-harmonic functions. Basic $H^p$ theory of such functions is developed: integral representations by measures and $L^p$ functions on $\mathbb T^n$, norm and weak$^\ast$ convergence at the distinguished boundary $\mathbb T^n$. Weak $(1,1)$-type estimate for a restricted non-tangential maximal function $M_{A, B}^{\mathrm{NT}}$ is derived. We show that slice functions $u(z_1, …, z_k, \zeta_{k+1}, …, \zeta_n)$, where some of the variables are fixed, belong in the appropriate space of separately $(\alpha', \beta')$-harmonic functions of $k$ variables. We prove a Fatou type theorem on a. e. existence of restricted non-tangential limits for these functions and a corresponding result for unrestricted limit at a point in $\mathbb T^n$. Our results extend earlier results for $(\alpha, \beta)$-harmonic functions in the disc and for $n$-harmonic functions in $\mathbb D^n$. Bibliography: 23 titles.

CONTINUOUS EXTENSIONS: THEIR CONSTRUCTION AND THEIR APPLICATION IN THE THEORY OF DIFFERENTIAL EQUATIONS

Homogeneous Polynomial Solutions to Constant Coefficient PDE's

Let F be a holomorphic function on an open set \(U \subseteq \mathbb{C}\). Write F = u+iv, where u and v are real-valued. The real part u satisfies a certain partial differential equation known as Laplace’s equation: $$\left( {\frac{{{{\partial }^{2}}}}{{\partial {{x}^{2}}}} + \frac{{{{\partial }^{2}}}}{{\partial {{y}^{2}}}}} \right)u = 0.$$ (7.1.1.1) (Of course the imaginary part y satisfies the same equation.) In this chapter we shall study systematically those C2 functions that satisfy this equation. They are called harmonic functions. (Note that we encountered some of these ideas already in §1.4.)