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1992 | Buch

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Harmonic Function Theory

verfasst von: Sheldon Axler, Paul Bourdon, Wade Ramey

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

Harmonic functions - the solutions of Laplace's equation - play a crucial role in many areas of mathematics, physics, and engineering. Avoiding the disorganization and inconsistent notation of other expositions, the authors approach the field from a more function-theoretic perspective, emphasizing techniques and results that will seem natural to mathematicians comfortable with complex function theory and harmonic analysis; prerequisites for the book are a solid foundation in real and complex analysis together with some basic results from functional analysis. Topics covered include: basic properties of harmonic functions defined on subsets of Rn, including Poisson integrals; properties bounded functions and positive functions, including Liouville's and Cauchy's theorems; the Kelvin transform; Spherical harmonics; hp theory on the unit ball and on half-spaces; harmonic Bergman spaces; the decomposition theorem; Laurent expansions and classification of isolated singularities; and boundary behavior. An appendix describes routines for use with MATHEMATICA to manipulate some of the expressions that arise in the study of harmonic functions.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Basic Properties of Harmonic Functions

Abstract

Harmonic functions, for us, live on open subsets of real Euclidean spaces. Throughout this book, n will denote a fixed positive integer greater than 1 and Ω will denote an open, non-empty subset of R ⁿ. A twice continuously differentiate, complex-valued function u defined on Ω is harmonic on Ω if

$$\Delta u \equiv 0 $$

where Δ = D ₁ ² + ... + D _n ² and D _j ² denotes the second partial derivative with respect to the j ^th coordinate variable. The operator Δ is called the Laplacian, and the equation Δu = 0 is called Laplace’s equation. We say that a function u defined on a (not necessarily open) set E ⊂ R ⁿ is harmonic on E if u can be extended to a function harmonic on an open set containing E.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 2. Bounded Harmonic Functions

Abstract

Liouville’s Theorem in complex analysis states that a bounded holo-morphic function on C is constant. A similar result holds for harmonic functions on R ⁿ. The simple proof given below is taken from Edward Nelson’s paper [7], which is one of the rare mathematics papers not containing a single mathematical symbol.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 3. Positive Harmonic Functions

Abstract

In Chapter 2 we proved that a bounded harmonic function on R ⁿ is constant. We now improve that result.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 4. The Kelvin Transform

Abstract

The Kelvin transform performs a role in harmonic function theory analogous to that played by the transformation f(z) ↦ f(1/z) in holomorphic function theory. For example, it transforms a function harmonic inside the unit sphere into a function harmonic outside the sphere. In this chapter, we introduce the Kelvin transform and use it to solve the Dirichlet problem for the exterior of the unit sphere and to obtain a reflection principle for harmonic functions. Later, we will use the Kelvin transform in the study of isolated singularities of harmonic functions.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 5. Spherical Harmonics

Abstract

From the theory of Fourier series on the unit circle we know that when n = 2, every f ∈ L ²(S) has an expansion of the form

$$f({e^{i\theta }})=\sum\limits_{m = - \infty }^\infty {{a_m}{e^{im\theta }},} $$

where the sum converges in L ²(S). In this chapter we will see that an analogous expansion is valid for functions f ∈ L ²(S) when n > 2, with objects known as spherical harmonics playing the roles of the exponentials e ^imθ.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 6. Harmonic Hardy Spaces

Abstract

In Chapter 1 we defined the Poisson integral of a function f ∈ C(S) to be the function P[f] defined on B by

$$P\left[ f \right](x) = \int_S {P\left( {x,\zeta } \right)f} \left( \zeta \right)d\sigma \left( \zeta \right)$$

(6.1)

.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 7. Harmonic Functions on Half-Spaces

Abstract

In this chapter we study harmonic functions defined on the upper half-space H. Harmonic function theory on H has a distinctly different flavor from that on B. One advantage of H over B is the dilation-invariance of H. We have already put this to good use in the section Limits Along Rays in Chapter 2. Some disadvantages: ∂H is not compact and Lebesgue measure on ∂H is not finite.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 8. Harmonic Bergman Spaces

Abstract

Throughout this chapter, p denotes a number satisfying 1 ≤ p < ∞. The Bergman space b ^p(Ω) is the set of harmonic functions u on Ω such that

$${\left\| u \right\|_p} = {\left( {\int_\Omega {{{\left| u \right|}^p}dV} } \right)^{1/p}} < \infty $$

.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 9. The Decomposition Theorem

Abstract

If K ⊂ Ω is compact and u is harmonic on Ω \ K, then u might be badly behaved near both ∂K and ∂Ω; see, for example, Theorem 11.18. In this chapter we will see that u is the sum of two harmonic functions, one extending harmonically across ∂K, the other extending harmonically across ∂Ω. More precisely, u has a decomposition of the form

$$u = \nu + w$$

on Ω \ K, where v is harmonic on Ω and w is harmonic on R ⁿ \ K. Furthermore, there is a canonical choice for w that makes this decomposition unique.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 10. Annular Regions

Abstract

An annular region is a set of the form {x ∈ R ⁿ : r ₀ < ∈x ∈ < r ₁}; here r ₀ ∈ [0, ∞) and r ₁ ∈ (0, ∞]. Thus an annular region is the region between two concentric spheres, or is a punctured ball, or is the complement of a closed ball, or is R ⁿ \ {0}.

Sheldon Axler, Paul Bourdon, Wade Ramey

Chapter 11. The Dirichlet Problem and Boundary Behavior

Abstract

In this chapter we construct harmonic functions on Ω that behave in a prescribed manner near ∂Ω. Here we are interested in general domains Ω ⊂ Rⁿ; the techniques we developed for the special domains B and H will thus not be available. Most of this chapter will concern the Dirichlet problem. In the last section, however, we will study a different kind of boundary behavior problem—the construction of harmonic functions on Ω that cannot be extended harmonically across any part of ∂Ω.

Sheldon Axler, Paul Bourdon, Wade Ramey

Backmatter

Titel: Harmonic Function Theory
verfasst von: Sheldon Axler
Paul Bourdon
Wade Ramey
Copyright-Jahr: 1992
Verlag: Springer New York
Electronic ISBN: 978-0-387-21527-3
Print ISBN: 978-1-4899-1186-5
DOI: https://doi.org/10.1007/b97238